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Full text of "Algebra I, Teacher’s Edition"

CK-12 Foundation 



CK-12 Algebra I Teacher's 

Edition 



Say Thanks to the Authors 

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Fay-Zenk McFarland 



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CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook mate- 
rials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative 
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Complete terms can be found at http://www.ckl2.org/terms. 

Printed: August 9, 2011 



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Authors 

Mary Fay-Zenk, Andrew McFarland 



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Contents 



1 TE Equations and Functions 1 

1.1 Variable Expressions 2 

1.2 Order of Operations 4 

1.3 Patterns and Equations 5 

1.4 Equations and Inequalities 7 

1.5 Functions as Rules and Tables 8 

1.6 Functions as Graphs 10 

1.7 Problem-Solving Plan 11 

1.8 Problem-Solving Strategies: Make a Table; Look for a Pattern 12 

2 TE Real Numbers 14 

2.1 Integers and Rational Numbers 15 

2.2 Addition of Rational Numbers 17 

2.3 Subtraction of Rational Numbers 19 

2.4 Multiplication of Rational Numbers 21 

2.5 The Distributive Property 23 

2.6 Division of Rational Numbers 25 

2.7 Square Roots and Real Numbers 26 

2.8 Problem-Solving Strategies: Guess and Check; Work Backward 28 

3 TE Equations of Lines 29 

3.1 One-Step Equations 31 

3.2 Two-Step Equations 32 

3.3 Multi-Step Equations 34 

3.4 Equations with Variables on Both Sides 35 

3.5 Ratios and Proportions 36 

3.6 Scale and Indirect Measurement 38 

3.7 Percent Problems 39 

3.8 Problem-Solving Strategies: Use a Formula 42 

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4 TE Graphs of Equations and 

Functions 44 

4.1 The Coordinate Plane 45 

4.2 Graphs of Linear Equations 47 

4.3 Graphing Using Intercepts 48 

4.4 Slope and Rate of Change 49 

4.5 Graphs Using Slope-Intercept Form 50 

4.6 Direct Variation Models 51 

4.7 Linear Function Graphs 53 

4.8 Problem-Solving Strategies - Graphs 54 

5 TE Writing Linear Equations 56 

5.1 Linear Equations in Slope-Intercept Form 57 

5.2 Linear Equations in Point-Slope Form 58 

5.3 Linear Equations in Standard Form 59 

5.4 Equations of Parallel and Perpendicular Lines 60 

5.5 Fitting a Line to Data 62 

5.6 Predicting with Linear Models 63 

5.7 Problem Solving Strategies: Use a Linear Model 64 

6 TE Graphing Linear 

Inequalities 66 

6.1 Inequalities Using Addition and Subtraction 67 

6.2 Inequalities Using Multiplication and Division 69 

6.3 Multi-Step Inequalities 71 

6.4 Compound Inequalities 72 

6.5 Absolute Value Equations 75 

6.6 Absolute Value Inequalities 77 

6.7 Linear Inequalities in Two Variables 78 

7 TE Solving Systems of 

Equations and Inequalities 79 

7.1 Linear Systems by Graphing 80 

7.2 Solving Linear Systems by Substitution 82 

7.3 Solving Linear Systems by Elimination through Addition or Subtraction 84 

7.4 Solving Systems of Equations by Multiplication 85 

7.5 Special Types of Linear Systems 87 

7.6 Systems of Linear Inequalities 88 

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8 TE Exponential Functions 90 

8.1 Exponent Properties Involving Products 91 

8.2 Exponent Properties Involving Quotients 93 

8.3 Zero, Negative, and Fractional Exponents 95 

8.4 Scientific Notation 97 

8.5 Exponential Growth Functions 98 

8.6 Exponential Decay Functions 99 

8.7 Geometric Sequences and Exponential Functions 100 

8.8 Problem-Solving Strategies (reprise Make a Table; Look for a Pattern) 102 

9 TE Factoring Polynomials 103 

9.1 Addition and Subtraction of Polynomials 104 

9.2 Multiplication of Polynomials 108 

9.3 Special Products of Polynomials 109 

9.4 Polynomial Equations in Factored Form 110 

9.5 Factoring Quadratic Expressions 112 

9.6 Factoring Special Products 114 

9.7 Factoring Polynomials Completely 115 

10 TE Quadratic Equations and 

Quadratic Functions 116 

10.1 Graphs of Quadratic Functions 118 

10.2 Quadratic Equations by Graphing 120 

10.3 Quadratic Equations by Square Roots 121 

10.4 Quadratic Equations by Completing the Square 122 

10.5 Quadratic Equations by the Quadratic Formula 125 

10.6 The Discriminant 126 

10.7 Linear, Exponential and Quadratic Models 127 

10.8 Problem Solving Strategies: Choose a Function Model 128 

11 TE Algebra and Geometry 

Connections; Working with Data 130 

11.1 Graphs of Square Root Functions 131 

11.2 Radical Expressions 134 

11.3 Radical Equations 136 

11.4 The Pythagorean Theorem and Its Converse 138 

11.5 Distance and Midpoint Formulas 139 

11.6 Measures of Central Tendency and Dispersion 140 

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11.7 Stem-and-Leaf Plots and Histograms 143 

11.8 Box-and- Whisker Plots 144 

12 TE Rational Equations and 
Functions; Topics in 

Statistics 146 

12.1 Inverse Variation Models 147 

12.2 Graphs of Rational Functions 149 

12.3 Division of Polynomials 150 

12.4 Rational Expressions 151 

12.5 Multiplication and Division of Rational Expressions 152 

12.6 Addition and Subtraction of Rational Expressions 153 

12.7 Solutions of Rational Equations 155 

12.8 Surveys and Samples 156 



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Chapter 1 

TE Equations and Functions 



Overview 

Equations and Functions consists of eight lessons that introduce students to the language of algebra. 
Suggested Pacing: 

Variable Expressions - 1 hr 
Order of Operations - 1 hr 
Patterns and Equations - 1 - 2 hrs 
Equations and Inequalities - 1 - 2 hrs 
Functions as Rules and Tables - 0.5 hrs 
Functions as Graphs - 1 hr 
Problem-Solving Plan - 0.5 hr 
Problem-Solving Strategies: - 2 hrs 
Make a Table; Look for a Pattern 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

The problem-solving strategies presented in this chapter, Make a Table and Look for Patterns, are foun- 
dational techniques. Making a Table can help structure a student's ability to organize and clarify the data 
presented and teach the student to communicate more clearly. When teaching Look for a Pattern, ask 
questions such as: 

• Can you identify a pattern in the given examples that would let you extend the data? 

• Do you observe any pattern that applies to all the given examples? 

• Can you find a relationship or operation(s) that would allow you to predict another term? 

Alignment with the NCTM Process Standards 

Two promising practices, focused on the communication standards, can be effectively used with these 
strategies. The first is setting aside a time on a regular basis (i.e. once a week) to have students write 

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about the problem solving they have been doing (COM. 2). Present a daily warm-up problem which is well 
suited to the strategy being learned such as Look for a Pattern. Students work on one problem a day, 
first individually and then as a group with teacher leadership and summation. Each day the problem is 
discussed and solved, and many different points of view are shared in the process (COM.l, COM. 3). At 
the end of the week, students are asked to write about any one of the problems they did earlier in the 
week. This practice pushes students to develop logical thinking skills, to benefit from the classroom work 
that was shared earlier in the week, and to learn to communicate mathematical ideas clearly (COM. 4). It 
also gives students a choice; they only have to write about one of the problems, and it can be the problem 
that made the most sense to them. Oftentimes, unfortunately, we do not honor enough what makes sense 
to students; we expect them only to be able to follow the logic presented to them. We must give them 
experiences of "sense-making" as well (RP.l, RP.2). 

A second practice is posting student work, such as: 

• gallery walks where work in progress can be viewed 

• posters that highlight exceptionally well done solutions 

• student essays posted on the classroom wall 

What matters is that students' work is valued and shared (RP.3). If these pieces are large enough to be 
seen from a distance, this practice helps students to recall various approaches to solving problems (RP.4) 
and keeps their thinking "alive." 

• COM.l - Organize and consolidate their mathematical thinking through communication. 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• COM. 4 - Use the language of mathematics to express mathematical ideas precisely. 

• RP.l - Recognize reasoning and proof as fundamental aspects of mathematics. 

• RP.2 - Make and investigate mathematical conjectures. 

• RP.3 - Develop and evaluate mathematical arguments and proofs. 

• RP.4 - Select and use various types of reasoning and methods of proof. 

1.1 Variable Expressions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Evaluate algebraic expressions. 

• Evaluate algebraic expressions with exponents. 

Vocabulary 

Terms introduced in this lesson: 

algebra 

generalize 

variables 

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equations 

substitution (evaluation, "plugging-in") 

values 

solution 



Teaching Strategies and Tips 

Although the properties of real numbers are not covered until the chapter Real Numbers, some basic working 
assumptions must be made in this chapter. Addition and multiplication involving negative numbers, for 
example, will probably not be new to the majority of your students, but it is necessary nonetheless for 
the completion of the Review problems. A comprehensive assessment prior to the end of the first week of 
classes is suggested so that you can gauge how much time you will need to cover Equations and Functions 
and Real Numbers. 

Unknowns are used as placeholders in equations like 4x + 3 = x - 2 and formulas like A = 2nr. Using 
unknowns offers great advantages. It prevents having to solve problems "from scratch." They allow the 
general formulation of arithmetical laws such as a + b = b + a and a X b = bxa for all real numbers a and b. 
Variables are used as shorthand in functional relationships such as, "Let x represent the number of cricket 
chirps heard in a 15 minute interval. Then the temperature outside is given by the function T{x) = mx + b, 
where T represents the outside temperature." 

The difference between equations and expressions is important. Basically, one has an equals sign and the 
other one does not; equations can be solved and expressions are evaluated. It will help the students to see 
several examples side-by-side: 

Equations: -x + 2 = 7, x + 2y = x-l, x=7 

Expressions: -x + 2, 2x + 2y, nr 2 

Coefficients represent repeated addition and exponents represent repeated multiplication. 

The process of evaluation is also called substitution. It may be explained to the students that since the 
variable is standing in place of all values (or some unknown value), substitution is an occasion to replace 
the letter with a value. Concerning notation: It is important for students to understand that parentheses 
are at their disposal at all times because they make calculations more explicit. 

Several examples can be presented, each involving a different aspect of the evaluation process. For example: 



One instance of the variable: Evaluate -3x + 2 when x = -2 

Two or more instances of the variable: Evaluate -3x + 7 - x + 1 when x = -2 

Two or more variables: Evaluate -3x + ly + 1 when x = -2 and y = 4 

Evaluation involving negative numbers and fractions: Evaluate ^x + 7 when x = -4 



Examples from geometry and physics allow the students to see the use and application of substitution and 
even provide motivation in other directions. See lesson Examples 5 and 7. 

It is helpful to include words and phrases associated with functions in your lesson. Establishing familiarity 
with words and phrases like, "input and output," "...in for x," "out of the expression comes... ," "value," 
"...the expression evaluated at x," etc., will help provide students with a foundation for the upcoming lesson 
on functions and function notation. 

Word or story problems are important to establish right from the beginning. 

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Error Troubleshooting 

Teachers are advised not to allow notation to become an obstacle for their students from the beginning. 
Writing should be in-line and organized and maintain a logical flow. Students may misread their work 
if their writing is illegible. Z's can look like 2's, +'s like lower-case t's, tiny negatives disappear into the 
paper, other stray marks become minus signs, etc. 

Students often compress their answers on paper, especially when they are used to writing across instead 
of in a downward fashion. 3x-l = 8 = 3x = 9 = x = 3. The equals sign is being used in at least two ways: 
equality between quantities, and equivalence (or implication) between equations. 

When evaluating algebraic expressions, students can easily lose track of negatives. 

Evaluate 1 - x, where x = — 1. 

Evaluate 1 - x 2 , where x = — 1. 

Evaluate 1 - 2=2 where x = — 1. 

The first example involves a variable with a negative "out in front" and the value being substituted is 
negative. The second involves the vulnerable combination of negatives and exponents. The third involves 
negatives and fractions. As a check, ask your students if they can tell the difference between 

-l 2 and (-1) 2 

1.2 Order of Operations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Evaluate algebraic expressions with grouping symbols. 

• Evaluate algebraic expressions with fraction bars. 

• Evaluate algebraic expressions with a graphing calculator. 

Vocabulary 

Terms introduced in this lesson: 

order of operations 
PEMDAS 
grouping symbols 
parentheses 
nested parentheses 

Teaching Strategies and Tips 

As the lesson illustrates, a string of symbols is meaningless without an order to the operations established 
beforehand. The usual order of operations - evaluate expressions inside parentheses, exponents, multipli- 
cation and division from left to right before addition, and subtraction from left to right - is introduced 
alongside the options of grouping symbols available to students. 

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Examples in this lesson illustrate the different kinds of situations that can arise involving grouping symbols: 
expressions without parentheses; expressions with parentheses (and other grouping symbols); inserting 
parentheses manually that are otherwise not inherent in the expression; parentheses within parentheses 
(nested parentheses - grouping symbols several layers deep); working with a complicated-looking expression 
(working from inner grouping symbol to outer grouping symbol), thereby being convinced that by sticking 
to the order of operations, its simplification does not need to be difficult. 

Students also learn to consider fraction bars as grouping symbols. The fraction bar is an invisible bracket 
- the numerator and the denominator need to be simplified before proceeding. 

Consider evaluating the following with and without adding parentheses: 

-x + 3, when x = -1 

Teachers are advised to use caution when presenting the mnemonic PEMDAS (Parenthesis, Exponent, 
Multiplication, Division, Addition, Subtraction). Though it is a highly effective way to get students to 
memorize the order of operations, the horizontal nature of our writing is suggestive of having M performed 
before D, and A before S . As you know, it's not multiplication before division or addition before subtraction. 
Students have responded positively to the vertical schematic: 

P 

E 
M-D 

A-S 

Error Troubleshooting 

As discussed previously, there is a potential for error in the horizontally written mnemonic, PEMDAS. 
Students, especially those who have consciously or unconsciously shut out mathematics and refuse to 
participate in it in any meaningful way, will inevitably ignore the left-to-right precept and go with what is 
suggested: M. before D, and A before S . 

Students should be reminded occasionally that subtracting negatives is equivalent to adding positives. 

1.3 Patterns and Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write an equation. 

• Use a verbal model to write an equation. 

• Solve problems using equations. 

Vocabulary 

Terms introduced in this lesson: 
horizontal axis 

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vertical axis 

increases 

decreases 



Teaching Strategies and Tips 

The process of taking givens and translating them into mathematical statements by way of symbols is 
called modeling. Use the four-step problem-solving process to break down problems. 

Step 1: Extract the important information. 

Step 2: Translate into a mathematical equation. 

Step 3: Solve the equation. 

Step 4: Check the result. 

When it comes to providing a list of verbal cues and their mathematical translation, it is helpful to 
complement the lesson with a more comprehensive list: 

at least 

at most 

as much as 

no more than x 

increased by y 

x decreased by y 

sum of x and y 

total of x and y 

x plus y 

gain 

raise 

more 

increase of 

difference between x and y 

x minus y 

loss 

less 

fewer 

take away 

multiply x by y 

product of x andy 

x times y 

double x 

twice x 

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triple x 

percent of x 

fraction of x 

quotient of x andy 

divide x by y 

x divided by y 

per 

Part of the learning process consists of learning from mistakes. If students are told that their answers are 
wrong without good reasons or explanations, they will simply lose out on what could have been a learning 
experience. Consider the following subtle example. 

Students were asked to simplify the expression: 

(-x)(-x)(-x) 

One student gave the answer: 

(-x) 3 

The exponent is odd, (-x) 3 so it can be further simplified to -x 3 . The two expressions can be interpreted 
differently if order of operations is the focus: 

(-x) 3 means to cube the opposite of x 

-x 3 means to take the opposite of x cubed 

Error Troubleshooting 

Require students to check their work. Some beneficial questions are: Does the solution make sense? How 
does it compare with a quick estimate I could make? 

Some students will forget to follow the order of operations. Encourage students to read the problem 
carefully. Example 3 from this lesson demonstrates how to use a pattern as a tool to develop the required 
equation. 

1.4 Equations and Inequalities 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write equations and inequalities. 

• Check solutions to equations. 

• Check solutions to inequalities. 

• Solve real-world problems using an equation. 

Vocabulary 

Terms introduced in this lesson: 

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inequality 

constants 

greater than 

less than 

greater than or equal to 

less than or equal to 

not equal to 

Teaching Strategies and Tips 

This lesson can be viewed as a generalization of the previous lesson. The emphasis here is placed on 
defining the variable correctly, then translating the words in the problem into a mathematical expression 
that contains those variables, using an equal sign or an inequality. 

Checking solutions to inequalities is inherently the same as checking solutions to equations. The variable 
is similarly replaced by a value and should produce a true statement. 

The process of simplifying equations can be stated as follows: 

1. Evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions according 
to the usual order of operations. Make proper use of the associative and distributive laws. 

2. Combine like terms. This means adding or subtracting variables of the same type. 

3. Add, subtract, multiply by, or divide by any value (except 0) on both sides of the equation. 

The exception, of course, is when multiplying or dividing an inequality by a negative, reversing the in- 
equality symbol in order to obtain an equivalent inequality. It is useful to present examples to students in 
which the inequality symbol does not get reversed and others in which the symbol must be reversed. 



Error Troubleshooting 

Have students write out and identify the givens in problems. 

If variables other than x are used students need to be careful that Z's and 2's are not confused. 

Students have trouble labeling the variables correctly. Have them locate the key question in the exercise 
such as: What? How much? When? How long? Where? How far? 

Students should be cautioned to avoid the common mistake when fractions are involved; failing to divide 
the entire numerator will lead to incorrect results. For example: 

1.5 Functions as Rules and Tables 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify the domain and range of a function. 

• Make a table for a function. 

• Write a function rule. 

• Represent a real-world situation with a function. 

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Vocabulary 

Terms introduced in this lesson: 



function 

independent variable 

dependent variable 

input 

output 

domain 

range 

table 



Teaching Strategies and Tips 

This lesson introduces students to the concept of a function. Functions are relationships, associations, 
dependences, between two quantities. Independent and dependent variables are distinguished. Students 
realize that not all real numbers are permissible in the functions they write for problems. They learn to 
identify the domain and range of a function. Consider the level of detail in the explanation to Example 1 
from the SE. 

Tables provide students with organizational tools for dealing with functions. Students are asked to make 
tables for explicit functions. Conversely, students are asked to write a function rule or expression from a 
given table of values. 

Emphasis is placed on representing real-world situations with functions. Examples include, the price a 
person pays for phone service depending on the number of minutes used, and the number of amusement 
park rides someone goes on depending on the total amount of money paid. 

Note: teachers frequently must explain that there is no implied multiplication in the notation: f(x), read 
"/ of x". 

Some examples include: 



The cost of postage for a letter is a function of its weight. 

The amount of flour used when preparing a recipe is a function of the number of servings desired. 

How much you pay for gas at a station is a function of the number of gallons pumped. 



Emphasize representing real- world situations with functions by writing out the function in words first. For 
example, 

total cost = flat fee + hourly fee X number of hours 



Error Troubleshooting 

Function notation can be an obstacle for many students, primarily because they do not understand its 
purpose. Some students will try to multiply the / and the x. 

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1.6 Functions as Graphs 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph a function from a rule or table. 

• Write a function rule from a graph. 

• Analyze the graph of a real world situation. 

• Determine whether a relation is a function. 

Vocabulary 

Terms introduced in this lesson: 

coordinate plane 

Cartesian plane 

origin 

x-axis 

y-axis 

coordinate points 

quadrants 

vertical line test 

Teaching Strategies and Tips 

This lesson introduces students to the coordinate plane and graphing a function from a rule or table. 

Example 1: Some students will graph the ordered pairs incorrectly. Emphasize starting at the origin, 
moving left or right, moving up or down, and then plotting the given point. 

It is a rule that functions specify the relationship between the independent and the dependent variables 
in a problem. Students usually understand that they must use the value of the independent variable to 
get the values of the dependent variable, but they should also understand that functions can be expressed 
verbally (in words), symbolically (as an equation), numerically (as a table of values), and graphically (as 
a graph). 

When students are determining whether a relation is a function, they are essentially noticing that for each 
input there is exactly one output. This can take many forms, depending on how the function is presented: 
Two or more numbers in the dependent column of a table; a graph intersecting the same vertical line in at 
least two places; etc. Students ought to be able to state the definition of a function in their own words, and 
teachers perhaps ought to be able to do this in at least a few different ways. For example, each x-value has 
only one y-value assigned to it, a value in the range can belong to more than one element in the domain, 
etc. Being a function would mean that more than one person in a class can have the same height, but one 
person cannot have two or more heights. 

As stated in the lesson, it is wise to work with a lot of values when you start graphing, but as students 
learn about different types of functions they will find that they will only need a few points in the table of 
values. This will help prevent substitution and arithmetical errors. If the points from a table do not lie on 
a line (assuming linear functions), then the student knows that one of the calculations has an error in it. 
Plotting too few points won't catch these kinds of mistakes. 

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Furthermore, a table of values should include inputs from both sides of the number line - negatives, 
positives, and zero Sometimes it is evident that negative numbers cannot be used for the independent 
variable because of domain considerations. 

Error Troubleshooting 

Example 1: Have students place arrow heads («-»,$) above the points in an ordered pair to help them 
remember which direction to move when plotting the point. 

Functions cannot have more than one output per input, but two inputs can have the same input. Examples 
that clarify this will help the student understand the definition of a function. 

1.7 Problem-Solving Plan 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Make a plan to solve the problem. 

• Solve the problem and check the results. 

• Compare alternative approaches to solving the problem. 

• Solve real-world problems using a plan. 

Vocabulary 

Terms introduced in this lesson: 

"read and understand the problem" 

knowns 

unknowns 

"solve the problem" 

"check and interpret the answer" 

Teaching Strategies and Tips 

In this lesson, students are introduced to problem solving and a new plan that will be used throughout the 
book. 

Some common strategies that students will learn are listed here for easy reference: 

draw a diagram 
make a table 
look for patterns 
guess and check 
work backward 
use a formula 
read and make graphs 

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• write equations 

• use linear models 

• use dimensional analysis 

• use the right type of function for the situation 

It is helpful to advise students that when translating sentences into expressions and equations, they should 
not attempt to solve the entire problem all at once. Instead, students should read the problem "phrase by 
phrase"; translate one phrase, then move on to the next. 

Teachers can ask their students to reread the question, rephrase it, or even read it aloud. 

Estimation is a good habit to develop as it helps confirm the answers students obtain after solving a 
problem. 

Error Troubleshooting 

The following list includes possible sources of mistakes for students while solving an applied problem: 

Not listing the given information 

Not restating the question being asked in your own words 

Not selecting a variable to represent the unknown quantity 

Not clearly stating what the variable represents 

Not looking for possible patterns 

Not looking up a definition or formula 

A common mistake students make is checking their answers in the equations that they constructed instead 
of in the original wording of the problems. 

1.8 Problem-Solving Strategies: Make a Table; 
Look for a Pattern 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Develop and use the strategy: Make a Table. 

• Develop and use the strategy: Look for a Pattern. 

• Plan and compare alternative approaches to solving the problem. 

• Solve real-world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 

"make a table" 
"look for a pattern" 

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Teaching Strategies and Tips 

In this lesson students learn how to develop and use the methods: make a table and look for a pattern. 

Teaching Tip: Review key phrases with students such as: What? How much? When? How long? Where? 
How far? (and even Who? and Why?). If students' answers (with appropriate units) do not answer the 
key phrase in the problem, then the student needs to return to it and either back-substitute to obtain the 
correct answer or start over by labeling the variables differently to answer the key phrase. 

In Example 1, the key phrase is How much was the total bill? By distilling this question from the rest of 
the problem, a student understands what is unknown: the total bill (how much the pizza costs). At this 
point, a variable can clearly be defined. 

The strategy make a table enables the problem-solver to recognize patterns and relationships within nu- 
merical data organized in tables. Students sometimes need help with table headings. In Example 2, the 
units are weeks, minutes per day, and minutes per week. By extracting this information and laying it out 
on paper, a student can readily infer the column headings. 

Look for a pattern has the problem-solver create only a minimum of rows until a pattern can be established 
and used to obtain the answer. 

Error Troubleshooting 

The following list includes possible sources of mistakes for students while solving an applied problem: 

Not listing the given information 

Not restating the question being asked in your own words 

Not selecting a variable to represent the unknown quantity 

Not clearly stating what the variable represents 

Not looking for possible patterns 

Not looking up a definition or formula 



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Chapter 2 



TE Real Numbers 



Overview 

In this chapter, students solve real-world problems involving addition, subtraction, multiplication, and 
division of real numbers; make use of the number line, absolute value, and square roots; and solve equations 
through the application of additive inverses and reciprocals. In the final lesson, students learn the methods 
of guess and check and working backwards. 

Suggested Pacing: 



Integers and Rational Numbers - 1 hr 
Addition of Rational Numbers - 1 hr 
Subtraction of Rational Numbers - 1 hr 
Multiplication of Rational Numbers - 1 hr 
The Distributive Property - 1 hr 
Division of Rational Numbers - 1 hr 
Square Roots and Real Numbers - 1 hr 
Problem-Solving Strategies: 
Guess and Check; Work Backward - 1 hr 



If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 



Problem-Solving Strand for Mathematics 

The problem-solving strategies presented in this unit, Guess and Check and Work Backwards, invite stu- 
dents to use their mathematical intuition and common sense in making sense of everyday experiences with 
mathematics. 

The technique of Guess and Check formalizes what many students have been doing for years: making 
intelligent guesses and, often, coming up with the right answer. The technique Working Backwards lays 
the groundwork for solving multi-step equations and helps students understand the need for using opposite 
operations. 

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Aligning with the NCTM Process Standards 

In the NCTM process standards, the concepts of Connections and Representation receive special attention. 
In Guess and Check and Working Backwards, students recognize and use connections among mathematical 
ideas (CON.l), especially when encouraged to look for and think about patterns as they work through the 
processes. As they write equations that express the reality of the problem situation (R.2 and R.3), they 
use the language of mathematics to express mathematical ideas precisely (COM. 4). 

• COM. 4 - Use the language of mathematics to express mathematical ideas precisely. 

• CON.l - Recognize and use connections among mathematical ideas. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• RP.2 - Make and investigate mathematical conjectures. 

• RP.4 - Select and use various types of reasoning and methods of proof. 

• R.2 - Select, apply, and translate among mathematical representations to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

2.1 Integers and Rational Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph and compare integers. 

• Classify and order rational numbers. 

• Find opposites of numbers. 

• Find absolute values. 

• Compare fractions to determine which is bigger. 

Vocabulary 

Terms introduced in this lesson: 

integers 

whole numbers 

greatest 

least 

even/odd numbers 

rational numbers 

ratio 

numerator 

denominator 

proper fractions 

improper fractions 

equivalent fractions 

reducing 

simplest form 

lowest common denominator 

common denominator 

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opposites 
absolute value 
entire expression 

Teaching Strategies and Tips 

Through the use of the number line in Example 1, students learn that "greater than" and "farther to the 
right" are equivalent, and that "less than" and "farther to the left" are equivalent. 

Use Example 2 to show that odd numbers are two apart. The frog will never land on an even number 
because: 

• the frog begins at an odd number 

• the frog only makes a jump of 2 

Students get a visual reinforcement that odd numbers are always two hops apart, even though the number 
2 is even. This will help them understand later why two consecutive, odd numbers can be written as x and 

x + 2. 

In Example 4, ask students to divide the pie diagrams into equal parts according to the denominators of 
each fraction and to shade according to their numerators. Common denominators have not been introduced 
at this point. 

Use Example 5 to contrast Example 4 and motivate the process of finding common denominators. The 
problem of determining the larger of two, nearly equal fractions is a harder one. This task becomes 
straightforward once students learn to rewrite fractions as equivalent ones. Note the switch from "pies" in 
Example 4 to "pie rectangles" in Example 5. 

Determining the larger of two rational numbers may be expressed symbolically: Given | and |, each is 
equivalent to g| and || respectively, where the choice of common denominator will never need to be larger 
than bd. Therefore, if ad > cb, then the fraction | is larger. If ad < cb, then the fraction | is larger. 

Additional examples: 

• Which is greater jL or | ? 

Solution: Since 9 • 3 = 27 > 26 = 13 • 2, then ^ > §. 

• Which is greater 4- or -£ ? 

Hint: Convert to mixed-numbers and see Example 5. 

• Challenge: Without a calculator, arrange from least to greatest, ^p^g- 

Hint: Compare any two fractions. Compare the larger of these two with the third. 

A mirror placed at the origin, perpendicular to the number line, reflects each whole number the same 
distance into the mirror as the distance it measures from the mirror. This demonstration shows that every 
real number has an opposite, with the exception of zero. One of the more common questions teachers get 
from their students is whether zero is positive or negative; it is clear from the demonstration above that 
zero has no reflection and therefore zero cannot be either. Positives and negatives are reflections of each 
other. See Example 7. 

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Distance is briefly explained in terms of absolute value. They are covered in the chapter Graphing Linear 
Inequalities. 

Use Example 8 to show that absolute value expressions are grouping symbols and the order of operations 
applies when evaluating them - students must treat them like a parenthesis in that what's inside must be 
simplified first. 

Error Troubleshooting 

General Tip: Often, when students cancel all the factors in a numerator, they write an answer of 0, since 
nothing is left. For example: 

_6_ _ t-j 
30 " %■$■$ * 5 

When canceling repeated factors from the numerator and denominator of a fraction, remind students that 
a 1 remains. 

In Example 7e, urge students to apply the -1 to the whole expression since "opposite of" means multiplying 
an entire expression by -1. 

Example: 

• The opposite of —2x + 1 is 2x - 1 and not 2x + 1. 

General Tip: Parentheses are an organizational tool. Students are encouraged to put them around an 
expression if it helps them prevent the above mistake. 

Example: 

• Find the opposite of —2x + 1. 

Solution: 

Start by putting parentheses around the expression: —{—2x + 1) 

Use Example 8d to help students see that — | - 15| + 15 and in general, —\x\ + \x\. Overly confident students 
reciting that the absolute value is always positive can incorrectly simplify —\x\ as \x\ instead of | - x\ as \x\. 

2.2 Addition of Rational Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Add using a number line. 

• Add rational numbers. 

• Identify and apply properties of addition. 

• Solve real-world problems using addition of fractions. 

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Vocabulary 

Terms introduced in this lesson: 

adding a negative number 
LCD/LCM 

least common multiple 
equivalent fractions 
commutative property 
associative property 
additive identity 
additive properties 

Teaching Strategies and Tips 

Use Examples 1-3 to provide a visual frame of reference for adding real numbers. Eventually, students will 
need to add numbers without using the number line. 

General Tip: Move to the left on the number line when faced with a negative number or subtraction. 

Additional example: 

• Represent the sum -7-5 on the number line. 

Solution: 

Starting at -7, move 5 units to the left: -7 - 5 = -12 

Use Examples 4 and 5 to introduce LCD (lowest common denominator) and LCM (lowest common multi- 
ple). Although they have different meanings, the difference is subtle. 

Additional examples: 

. Find the LCM of 24 and 30. 

Solution: 

The lowest number that both 24 and 30 divide into without remainder is 120. 

• Simplify ^ + i. 

Solution: 

In order to combine these fractions we need to rewrite them over a common denominator. We are looking 
for the lowest common denominator (LCD). We need to first identify the lowest common multiple (LCM) 
of 24 and 30. 

5 | 1 _ 25 | 4 _ 29 
24 + 30~120 + 120~120 

Teachers are encouraged to reinforce the notions of LCD and LCM using visual representations as illustrated 
in Example 4. 

To help students learn the difference between the associative and commutative properties state the rule 
that is used when doing each example in the classroom and then have them note the differences. 

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Error Troubleshooting 



General Tip: Students who have difficulty with LCD and LCM at this point, might benefit from factor 
trees, which are discussed in the next lesson. 

For instance, in Example 6, students are asked to combine two denominators having a common factor. 
The factor trees for 12 and 9 are: 

. 12 = 2x2x3 
. 9=3x3 

To find the LCM, start by taking all of the prime numbers {2,2,3} of 12. Then ask what prime numbers 
of 9 are missing from this list? The second 3 is missing. Therefore, the LCM of 12 and 9 is the product of 
the numbers in the list {2,2,3,3} 

2x2x3x3 = 36 

General Tip: Students need constant reminding that adding fractions cannot take place without common 
denominators. Although memorizing rules such as, 

a I b a+b 



a I c 



ad+bc 



b ' d bd 

may prove helpful, teachers are encouraged to explain why denominators must be the same. Use an 
approach similar to the following: 

• Measurements involving feet and inches cannot be added until a common unit is chosen. For example, 
the sum of 7 inches and 3 feet is not equal to 10 inches, 10 feet, or 10 feet - inches. Converting 3 feet 
to 36 inches allows us to add the 7 inches to it: 7 in + 36 in = 43 in. 

• The same is true with fractions. "Halves" and "fourths" must be first converted to the common unit 
"fourths." Similarly, fractions representing "halves" and "thirds," will need to be converted to a third 
unit, "sixths," the LCM of 2 and 3. 

• Example 4 uses a visual argument to justify the need for common denominators. In one direction, 
the "cake" is sliced according to the first fraction, 3/5; in the other direction it is sliced according to 
the second fraction 1/6. The sum of 3/5 and 1/6 can now be found - but only in terms of the little 
squares since the total number of squares represent the common unit. 

2.3 Subtraction of Rational Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find additive inverses. 

• Subtract rational numbers. 

• Evaluate change using a variable expression. 

• Solve real world problems using fractions. 

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Vocabulary 

Terms introduced in this lesson: 

additive inverse 

opposite 

opposite/inverse process 

subtraction 

method of prime factors 

simplify 

simplest form 

invisible denominator 

speed 

difference 

change 

Teaching Strategies and Tips 

General Tip: Additive inverses provide a way to subtract two real numbers. Students learn that the 
operation of subtraction can be replaced with addition by adding the additive inverse of the number. 

Subtraction by adding additive inverses applies as well to fractions. 

Example: 



3_6_3 / 6\ _ 3 +(-6) _ -3 

7 ~ 7 ~ 7 + w) ~ 7 ~ y 



Although it may be easier for students to keep it as subtraction: 

3 _ 6 _ 3-6 _ 3 + (-6) _ -3 
7 ~ 7 ~ 7 ~ 7 ~ T 

Tips on factor trees: 

• Do not include the trivial factors of 1 in the branches of factor trees. 

• The first two branches of the factor tree can be any pair of factors. 

• For instance, in Example 2, 90 = 9x10 and 126 = 9x14. Students may choose differently: 90 = 45x2 
and 126 = 21 x 6. 

• To find the LCM of three numbers, no more than three factor trees need to be constructed. In 
general, there will be as many factor trees as numbers whose LCM is being sought. See Example 4 
for three numbers. 

In Example 4, the "invisible denominator" is a useful way of getting students to realize that 

x 
X= l 

That is, integers are rational numbers. Every whole number can be viewed as a rational number whose 
denominator is 1. 

Use Examples 5 and 6 to demonstrate the concept of change. 
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• A variable expression is used to evaluate change in speed and change in light intensity, respectively. 
The functions are provided already; and therefore do not need to be derived from scratch. 

• The focus is on the concept of change, which will be new for many students. Exploration is encouraged 
through tables and graphs. 

• To find the change in a quantity using a function, cast this as a subtraction problem: subtract first 
quantity from second quantity. 

• Explaining the results in words will benefit students. Positive change in speed means an increase in 
speed. The change is negative in Example 6: the intensity reduces as the train travels farther away 
from the light source. 

Error Troubleshooting 

General Tip: For many students, the combination of subtraction and finding common denominators is 
prone to errors. Practice is crucial. See Examples 3 and 4 as well as Review Questions la - li. 

Example 5 and 6: Change = second quantity - first quantity; i.e., subtract the first quantity from the 
second quantity. In context: 

• Change = Speed 2 - Speed 1 

• Change = Intensity 2 - Intensity 1 

2.4 Multiplication of Rational Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

. Multiply by -1. 

• Multiply rational numbers. 

• Identify and apply properties of multiplication. 

• Solve real-world problems using multiplication. 

Vocabulary 

Terms introduced in this lesson: 

argument 

multiplicative properties 
multiplicative identity property 
distributive property 

Teaching Strategies and Tips 

Changing the sign of a number is equivalent to multiplying it by -1. In Example 1, students find the oppo- 
site of several numbers and expressions being careful to use parentheses when appropriate and simplifying 
the result. 

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Use Examples 2 and 3 to justify why, when multiplying two fractions, the numerators multiply together 
and the denominators multiply together. 

Use the product of three or more fractions as in Examples 4c and 4d as an extension of the multiplication 
rule. 

Additional examples: 

• Multiply the following rational numbers. 

3 5 

IT' 7 

Hint: The product of two rational numbers is the product of their numerators divided by the product of 
their denominators. 

• Multiply the following rational numbers. 

11 M 
I'I'I 

Hint: Multiply all the numerators and all the denominators. Do not covert the improper fraction to mixed 
form. 



Multiply the following rational numbers. 



5.12 

7 



Hint: Rewrite the 12 as 12/1, using the "invisible 1". 

Students first learn about the convenience of canceling before multiplying in Examples 4d and 5. 

Additional example: 

• Multiply the following rational numbers. 

24 8 9 
33 ' 27 ' 64 

Solution: 

24 8 9 3-8 8 9 _ %•$ $ _ 1 
33 ' 27 ' 64 ~ 3-11 ' 3~9 ' 8~8 ~ #-11 ' 3~0 ' jTj* ~ 33 

Use Examples 6-8 to introduce the four properties of real numbers which involve multiplication: the 
commutative, associative, multiplicative identity, and distributive properties. 

• A geometric interpretation of the commutative property is to consider finding the area of a rectangle. 
Lx W is the same number no matter how you draw the rectangle or what you call L and W; therefore, 
L X W = W XL. Similarly, the commutative property says that the order for multiplying any two real 
numbers does not matter. See Example 6. 

• The associative property of multiplication concerns three or more numbers. Just as for addition, the 
sum is the same regardless of how they are grouped and in which pair the multiplication takes place 
first. 

• State the rule being used in each example you do in the classroom. 

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Error Troubleshooting 

Example lb: The opposite of n is simply -1 • (n) = —n. There is no need to use the decimal expansion. 

Example lc: Multiply both terms of the expression by -1. This will make more sense to students after 
covering the distributive law in the next lesson. 

Additional example: 

• Find the opposite of the expression 

x - 4y + 1 

Hint: multiply each of the three terms by -1. 

The difference between absolute value and other grouping symbols is that multiplying absolute value by 
-1 will not affect the argument; that is, a negative will not distribute into the absolute value. See Example 
Id. 

General Tip: It is helpful to note that 

• \x\ and | - x\ are always positive 

• —\x\ is always negative. 

General Tip: A common mistake is to forget to cancel like factors before multiplying the fractions, as the 
numbers will only get larger and thus harder to factor. Have students factor numerators and denominators 
first to remove any repetitions by canceling. Then carry out the remaining easier multiplication. 

2.5 The Distributive Property 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Apply the distributive property. 

• Identify parts of an expression. 

• Solve real-world problems using the distributive property. 

Vocabulary 

Terms introduced in this lesson: 
distributive property 

Teaching Strategies and Tips 

In the lesson introduction: 

• Students see that distributing gift bags among the children is equivalent to distributing numbers. 

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• Care must be taken in using the abbreviations p,f, and c for photo, favor, and candy, respectively. 
These are not variables but units of measurement. Students can misconstrue the notation and 
intention - these quantities are not unknown and do not need to be solved for. 

Use Examples 1 and 2 to show that some expressions can be simplified in more than one way. Use the 
distributive property and apply the order of operations to obtain the same result. 

Additional examples: 

• Use the distributive property to simplify: 

-4(5 - 7) 

Solution: -4(5 - 7) = -4 • 5 + (-4) • (-7) = -20 + 28 = 8 
Note: Use care with the negatives. 

• Use order of operations to simplify: 

-4(5 - 7) 

Solution: -4(5 - 7) = -4(-2) = 8 

In Examples 2 and 3b- 3d, have students deal with the minus sign by committing to (1) using additive 
inverses, or (2) subtraction, but not both. For instance, Example 3b can be solved in two ways: 

1. 7(3x - 5) = 7(3x + (-5)) = 7(3x) + 7(-5) = 21x - 35 

2. 7(3* - 5) = 7(3*) - 7 • 5 = 21x - 35 

Use Example 4 to demonstrate the hidden distributive property. Fractions with two or more terms in their 
numerators can be simplified using the distributive property. The hidden distributive property is based on 
the rule for multiplication of rational numbers: 

a 1 

Point out in Example 5 that the distributive property equally applies on the right. Steel required = (4 + 5)8 
is also a correct answer. 

Additional example: 

• Simplify using the distributive property. 



(3x-l)2 

Solution: (3x - 1)2 = (3x)2 - 1 • 2 = 6x - 2 
In Example 6, 

• Some students will want to round up rather than down. Have students complete the problem by 
rounding up to see that they would not have enough money. 

• Point out that the distributive property equally applies to factors with three or more terms. 

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Error Troubleshooting 

In Example 3d, the x's will eventually cancel. Students can forget to reduce in the end. 
The 3 in Example 4b does not cancel. This is a common mistake made by students. 
Additional example: 



Simplify the following expression. 

ax — 2 fix — 2 



x-2 



Solution: — — can be rewritten as -{ax — 2) Distribute the -: 

1, ax 2 2 

— {ax — 2) = = x 

a a a a 

2.6 Division of Rational Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find multiplicative inverses. 

• Divide rational numbers. 

• Solve real-world problems using division. 

Vocabulary 

Terms introduced in this lesson: 

multiplicative inverse 

reciprocals 

invert the fraction 

improper fraction 

invisible denominator 

speed 

distance 

time 

Teaching Strategies and Tips 

Draw an analogy between the division and subtraction of rational numbers. 

• A subtraction problem can be recast as an addition problem using additive inverses (opposites). A 
division problem can be recast as a multiplication problem using multiplicative inverses (reciprocals). 

• When a number is added to its opposite, the additive identity, 0, is obtained. When a number is 
multiplied by its reciprocal, the multiplicative identity, 1, is obtained. 

In Example lc, remind students that a mixed number needs to be converted to an improper fraction before 
determining the multiplicative inverse. 

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Error Troubleshooting 

In Example Id, point out that finding the multiplicative inverse of the expression will not affect the 
negative. See also Example 2d. 

• The reciprocal of — ^ is —-. (Invert the fraction.) 

• The opposite of -* is -.(Multiply by -1.) 

2.7 Square Roots and Real Numbers 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find square roots. 

• Approximate square roots. 

• Identify irrational numbers. 

• Classify real numbers. 

• Graph and order real numbers. 

Vocabulary 

Terms introduced in this lesson: 

square root 
principal square root 
positive square root 
radicals 

perfect squares 
prime factors 
rational number 
irrational number 
approximate answer 
approximation 
non-repeating decimals 
simplest form 
integer 
sub-intervals 

Teaching Strategies and Tips 

Contrary to what students think, there is no discrepancy between there being two possible values for b, 
given a positive x, such that b 2 = x and only one value for yfx. The positive number b is the principal 
square root. It is the value of the function. 

Point out that y/~ and M mean the same thing. The "2" is understood. 

In Example 1: 

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• Use factor trees to break down the radicands into as many perfect squares as possible. 

• Primes which appear an even number of times constitute a perfect square. 

• Any unpaired factors are left under the radical sign. 

• The convention is to leave any irreducible radical in the form: (square root) -^/(irreducible part) 

Lots of practice early on makes it easier for students when the radicals become more complicated and 
involve variable expressions. 

Use Example 3 to multiply, divide, and simplify radical expressions. State which rules were used in 
classroom examples. 

In Example 4, students use a calculator and round their answer to three decimal places. Review rounding 
decimals. 

Motivate irrational numbers in Example 5. 

• Irrational numbers complete the set of real numbers. 

• They cannot be expressed as ratio of two integers. 

• They have an unending (non-terminating), seemingly random (non-repeating) decimal expansion. 

• Some irrational numbers: it, -, y2, V3, V an y prime 

In Example 5, students identify the given numbers. A number is rational if it 

• can be expressed as a fraction of integers. 

• has a finite decimal expansion. 

• has a repeating block of digits in its decimal expansion. 

Have students check their calculator displays for repeating blocks of digits, not just patterns. For example, 
the two numbers below have obvious patterns, but no repeating blocks, and therefore are irrational: 

. 0.010010001000010000010000001 . . . 

. 0.12345678910111213141516171819202122 . . . 

Remind students that integers can be written as fractions with a 1 in the denominator. See Example 6a 
and 6b. The strategy in Example 6e is to simplify first. 

The strategy in Example 8 is to use a calculator to find the decimal expansions of the numbers rounded 
to as many places as is needed to identify each. Since all the numbers are between 3.1 and 3.2, going out 
to three decimal places is sufficient. 

Error Troubleshooting 

In Problem 3 of the Review Questions: 

• Students should find repeating blocks of digits before claiming that a number is rational. 

• It is not sufficient to claim that the "unpredictable" decimals of a number on a calculator display 
belong to an irrational number. The decimal expansion of 2/19 has 18 seemingly random digits in its 
repeating block; therefore, the rational number 2/19 would go "unchecked" on an ordinary calculator 
display because it could not show enough digits. 

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2.8 Problem-Solving Strategies: Guess and Check; 
Work Backward 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Develop and use the strategy: Guess and Check. 

• Develop and use the strategy: Work Backward. 

• Plan and compare alternative approaches to solving problems. 

• Solve real-world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 

guess and check 
working backwards 

Teaching Strategies and Tips 

Use Example 1 to introduce guess and check. Teachers are encouraged to postpone solutions involving 
systems of equations (two variables) until chapter Solving Systems of Equations and Inequalities. 

Allow students to strategize from their guesses. The guessing process will often lead to unexpected patterns 
that can serve to make better guesses along the way: 

• In Example 2, one guess yields a sum of 24, which is half of the desired 48; therefore, the initial 
numbers should be multiplied by 2. 

• In Example 4, note the pattern given by the relation: when Nadia's age is decreased by 1, her father's 
age decreases by 4. This observation leads to the answer. 

• In Example 6, allow students to keep guessing until the total costs are the same. Students may notice 
however that for an increase of 10, the difference between total costs falls by $1. 

Use Example 3 to show how to work backward. Reverse the steps starting with the result until the unknown 
is obtained. 

General Tip: Teachers are encouraged to compare alternative approaches to some of the problems. 

Error Troubleshooting 

General Tip: Remind students to check their work. Ask: Does the answer make sense? 



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Chapter 3 

TE Equations of Lines 



Overview 

Rules for solving one-step, two-step, and multi-step equations, and equations with variables on both sides 
are presented. By the end of the fourth lesson, students solve applied problems involving general linear 
equations. Students reason with ratios and per cents, construct proportions and apply them to scaled 
drawings, and use formulas as a problem-solving strategy. 

Suggested Pacing: 

One-Step Equations - 1 hr 

Two-Step Equations - 1 hr 

Multi-Step Equations - 1 hr 

Equations with Variables on Both Sides - 1 hr 

Ratios and Proportions - 1 hr 

Scale and Indirect Measurement - 1 hr 

Percent Problems - 1 hr 

Problem-Solving Strategies: Use a Formula - 1 hr 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

Use a Formula is the problem-solving strategy highlighted in this unit. Many teachers and students 
believe that knowing what formula to use or apply to a given problem turns what may have been a difficult 
"problem" into more of a straightforward exercise. Substituting in known quantities and using a formula, 
they reason, is simple. 

For some students, however, following the steps of a formula is not a simple process, especially if there's 
an unknown quantity in the formula. Still for others, using a formula with unfamiliar terms such as ohms 
or finding the principal using a formula such as / = Pr t , is simply baffling. The first case, of course, 
is an ideal time to encourage students to apply a previously practiced skill like Working Backwards. By 
using opposite operations to "undo" an equation, students acquire a new formula, one they have generated 
themselves. 

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When presenting formulas to students, an excellent practice is to help students understand where the 
formula comes from in the first place. A straightforward example of this would be the formula P = 2/ + 2w. 
Students can easily understand that this formula for the perimeter of a rectangle matches the physical 
process of "marching around" the figure in fact or in their mind. Similarly, a mental image of a grid on 
which a rectangle is outlined verifies why the area formula for a rectangle is A = Iw. 

The challenge for us as teachers is to relate as many new formulas as possible to formulas that have already 
become self-evident to the students: 

• One classic strategy is to encourage students to create a visual image of the area of a parallelogram 
in relation to a rectangle, which can enclose the parallelogram when a triangular portion is shifted 
left or right, and which, therefore, exhibits an equivalent area. 

• Another strategy is to illustrate the formula for the area of a circle by cutting a circle into pie-slice 
wedges and rearranging them into a near-parallelogram, which gives students real reason to believe 
that the area of a circle really is "pi times radius squared." (This particular model makes an excellent 
poster for the classroom and can be an even more effective poster when it is the well-executed work 
of a classmate.) 

• Yet another strategy is to demonstrate how percentages work in retail stores by using mark-ups and 
discounts as examples. Having different groups of students compute two different scenarios - such 
as those presented in Example 14 of the Percent Lesson - can be an easy and convincing way for 
students to understand the algebraic mathematics involved in such situations. 

Helping students make sense of mathematical formulas can teach them to "trust" the formulas they are 
asked to use and cannot yet prove (such as Ohm's Law which relates volts, amps, and resistance). 

Aligning with the NCTM Process Standards 

The NCTM Process Standards that are addressed in the use and understanding of formulas include: 
connections, representation, and reasoning and proof. The connections between mathematics and science 
become readily apparent in this unit as students recognize and apply mathematics in contexts outside 
of mathematics (CON. 3) - e.g. questions about the speed of sound, resistance, voltage, and the time 
required for a jet plane to climb a given distance. Several formulas illustrate how business and finance use 
mathematics to relate costs, show profit and loss margins, and calculate interest rates. These formulas 
also encourage students to recognize and use connections among mathematical ideas in the use of opposite 
operations to solve problems (CON.l). 

As students strive to understand the derivation of various formulas, many connections occur (CON. 2). Stu- 
dents use representations repeatedly to model and interpret physical, social and mathematical phenomena 
(R.3). Conjectures are made and investigated (RP.2), arguments are developed and evaluated (RP.3), and 
various types of reasoning and methods of proof (RP.4) are utilized. 

• CON.l - Recognize and use connections among mathematical ideas 

• CON. 2 - Understand how mathematical ideas interconnect and build on one another to produce a 
coherent whole 

• CON. 3 - Recognize and apply mathematics in contexts outside of mathematics 

• RP.2 - Make and investigate mathematical conjectures 

• RP.3 - Develop and evaluate mathematical arguments and proofs 

• RP.4 - Select and use various types of reasoning and methods of proof 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena 

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3.1 One-Step Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve an equation using addition. 

• Solve an equation using subtraction. 

• Solve an equation using multiplication. 

• Solve an equation using division. 

Vocabulary 

Terms introduced in this lesson: 

equal 

equation 

isolate 

linear equation 

Teaching Strategies and Tips 

Use the introductory problem to motivate equivalent equations, since it can be solved two ways: 

• The cost plus the change received is equal to the amount paid, x + 22 = 100. 

• The cost is equal to the difference between the amount paid and the change, x = 100 - 22 

Examples 1 and 2 are essentially the same; constant terms must be added to both sides to isolate x on the 
left. Adding vertically can benefit students. 

Additional Example. 

Solve 12 = -4 + x. 

Solution. To isolate x, add 4 to both sides of the equation. Add vertically. 

12 = -^ + x 

+4 = +£ 
16 = x 

The variable in Example 3 is not the usual x. Remind students that the letter of the variable does not 
matter. Additional Example. 

Solve -21 = n + 14. 

Hint: To isolate n, subtract 14 from both sides of the equation. 

In Examples 4-6, teachers may opt to solve the equations by adding the opposite in lieu of subtracting. 

Example. 

Solve -17 = x + 8. 

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Hint: To isolate x, add -8 to both sides of the equation. Add vertically. 

-17 = x + $ 

-8 = -$ 
-25 = x 

Use Example 6 as an example of an equation with fractions. Remind students to find common denomina- 
tors. 



Point out in Example 8 that in general, 



ax a 
~b = b X 



which will help students isolate x in one step, multiplying by the reciprocal of a/b. 
Note that the equation in Example 10 can be written in dollars or in cents: 

• 5x = 3.25 (x in dollars) 

• 5x = 325 (x in cents) 

Although Examples 13 and 15 can be solved by making a table and Example 14 by guessing and checking, 
teachers are encouraged to help students setup and solve an equation of the type presented in the lesson. 

Error Troubleshooting 

General Tip: After the constant term is canceled in a one-step equation, the variable must be carried down 
onto the next line. Remind students to write the x =. 

General Tip: Students forget to perform the same operation on both sides of an equation. Have students 
use a colored pencil to write what they are doing to both sides of the equation. 

3.2 Two-Step Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a two step equation using addition, subtraction, multiplication, and division. 

• Solve a two-step equation by combining like terms. 

• Solve real-world problems using two-step equations. 

Vocabulary 

Terms introduced in this lesson: 

two-step equations 

like terms 

combining like terms 

unknown 

equation in two variables 

www.ckl2.org 32 



Teaching Strategies and Tips 

Use Example 1 to motivate the process of solving two-step equations: 

• Two marbles can be removed from each pan first. 

• The first step in solving two-step equations is to move the constant away from the variable term. 

• Whereas it was not possible before, the marbles can now be divided into three groups. 

• The second step in solving two-step equations is to isolate the variable by dividing by its coefficient. 

• It is a small jump for students to write an algebraic expression based on the equality implied by the 
pans and solve it in an analogous way. 

To keep the pans in equilibrium, Example 1 also teaches that, "what is done to one side must be done to 
the other side". 

Because the solution to Example 2 is negative, the balance strategy of Example 1 will not apply. Use a 
similar problem in which the solution is positive to demonstrate the balance strategy for variables buried 
in parentheses. 

Example. 

Six bags each containing the same unknown number of blue marbles and 1 red marble are placed on one 
side of a balance. 12 red marbles are put on the other side. The scales balance. How many blue marbles 
are in each bag? Assume the marbles weigh the same and the bags weigh nothing. 

Solution. The unknown quantity is the number of blue marbles in each bag and is denoted by x. The 
problem can be summed up as "Six bags of x blue marbles and 1 red equals 12 red marbles" and translated 

as 

6(x+l) = 12 

This is an example of an equation where x is buried in parentheses. To find the number of blue marbles, 
proceed in one of two ways: (1) Observe that 1 bag weighs the same as 2 red marbles. This is equivalent 
to dividing both sides of the above equation by 6. (2) Empty the contents of each bag onto the pan. There 
will be 6 red marbles and 6 times the number of blue marbles that was in one bag. This is equivalent to 
distributing the 6 in the above equation. In both approaches, the equations are reduced to the familiar 
one-step and two-step equations, respectively. 

The first approach is easier since 12 is evenly divided by 6. 

0(jc+1) _ 12 
6 

x + X = 2 

-X = -i 

x= 1 
General Tip: The first step in solving equations with variables buried in parentheses depends on: 

• Whether the constant is evenly divisible by the coefficient. See Example 2. 

• Whether fractions are present. See Examples 3 and 4. 

Warm-up to Examples 5 and 6 with exercises similar to the following: 
Which of the following pairs of expressions are like terms? 

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• x and 5x (Yes.) 

• x and xy (No.) 

• x and x 2 (No.) 

. -11 and 11 (Yes.) 

In Examples 8 and 9, two- variable equations will result. Students substitute one of the givens for one 
variable to determine the other. 

Error Troubleshooting 

Watch for the switch from Example 9(h) to 9(iii), from Celsius to Fahrenheit. 

Review Question lc. Remind students to distribute the negative to both terms in the parentheses. 

Review Question lg. Hint: Write the s term with a common denominator as |s. 

3.3 Multi-Step Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a multi-step equation by combining like terms events. 

• Solve a multi-step equation using the distributive property. 

• Solve real-world problems using multi-step equations. 

Vocabulary 

Terms introduced in this lesson: 
multi-step equation 

Teaching Strategies and Tips 

Use this lesson to introduce equations requiring several steps to isolate the variable. Although the examples 
and exercises are more involved than the previous two lessons, there are no new procedures to learn. 

An alternative solution to Example 1 is to have students multiply both sides of the equation by the LCD 
to clear the fraction first. 

Example 2 contains an x buried in parentheses. Possible questions to ask students about the way to proceed: 

• Do you want to avoid a fraction if possible? 

• Does distributing first amount to fewer steps? 

As students work through more examples, they will recognize the shorter solutions. As an informal rule, 
distribute when fractions will result from dividing; otherwise divide. 

• Example: 3(1 - x) = 7. Dividing both sides by 3 results in a fraction, since 7 is not a multiple of 3. 
Since fractions are to be avoided, distribute. 

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• Example: 3(1 - x) = 12. Since 12 is a multiple of 3, divide. Distributing will result in more steps. 

• Example: 1(1 — x) = 7. Fewer fractions result by multiplying both sides of the equation by the 
reciprocal of 3/2. Alternatively, clear the fraction by multiplying both sides of the equation by 2. 

• Example: 3(1 - x) = jj. To avoid having to divide the fraction by 3, distribute first. Alternatively, 
clear the fraction by multiplying both sides of the equation by 11. 

• Example: |(1 — x) = |. First clear the fractions by multiplying both sides of the equation by the 
LCD. 

Error Troubleshooting 

In Example 3, remind students to distribute the negative along with the 7; i.e., 

4(3x - 4) - 7(2x + 3) = 3 

=> 4(3x - 4) + (-7) (2* + 3) = 3 

=» 12x - 16 - Ux - 21 = 3 

Each set of parentheses needs to be expanded before combining like terms in Examples 3 and 4. 

The equation in Example 4 contains both fractions and decimals. Combining like terms is easier by 
converting the fractions to decimals first. Decimals do not require a common denominator! Alternatively, 
clear the fractions by multiplying both sides of the equation by the LCD, 10. The remaining integers and 
decimals can be combined. 

3.4 Equations with Variables on Both Sides 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve an equation with variables on both sides. 

• Solve an equation with grouping symbols. 

• Solve real-world problems using equations with variables on both sides. 

Vocabulary 

Terms introduced in this lesson: 

collect like terms 
rational function 

Teaching Strategies and Tips 

In Examples 1 and 2, use the balance analogy to motivate solving equations with variables on both sides. 
It makes no difference in the answer whether the variable or the constant are moved first in an equation. 

• In Example 3, students learn to move the variable term and not the constant. This results in the 
fewest number of steps. 

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• Whether the final step in solving an equation gives "x =" or "= x", the answers will be the same. 

• Some teachers may prefer to have students commit to collecting like variable terms on the left of the 
equal sign and the constant terms on the right for the first several problems. 

• Other teachers will encourage students to isolate the variable on the side that is most convenient. 

• Essentially students should be seeking the quickest routes to the isolated variable. 

• As a class, have students vote on how to solve the equations in the Review Questions. 

In Examples 5 and 6, the fraction must be eliminated first by multiplying both sides by the LCD. Then 
the distribution can take place. Any other way means extra steps. 

Additional Example. 

Solve 5+ f = ZgS-l. 

Solution. Start by eliminating the fractions; multiply by the LCD. 

30 + 4x = 7-x-6 

Reminder 1: Multiply each term by the LCD. Do not neglect the -1 on the right or the 5 on the left. 

Reminder 2: The result of multiplying -# by the LCD is 4x because -£ • 6 = -r{2 •$) = -j- ■ 2 = 4x. Do not 
neglect any "un- cancelled" factors. 

Example 8 consists of many givens. Have students organize the given information before translating into 
an equation to solve. 

Error Troubleshooting 

Use Example 7 to show that the whole factor (x + 3) is the LCD in the equation. Warn students against 
separating the two terms in the denominator; i.e., 

14* 



(x + 3) 

Note that Example 7 briefly introduces rational equations. After eliminating the equation of the rational 
function, it becomes an equation they know how to solve. 

3.5 Ratios and Proportions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write and understand a ratio. 

• Write and solve a proportion. 

• Solve proportions using cross products. 

Vocabulary 

Terms introduced in this lesson: 

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ratio 
unit rate 
proportion 
cross multiplying 

Teaching Strategies and Tips 

Use the introductory problem and Example 1 to motivate ratios. 

• Have students compare ratios for value and number of coins in the introductory problem; and the 
difference in price and the ratio of prices in Example 1. 

• Ask follow-up questions such as: 

Which ratio is more useful? 

Which ratio does the better job summarizing the situation? 

How would you explain to the boy that he is actually getting a bad deal? 

Can you find an example of each ratio in daily life? 

In Example 1, which measure of comparison is more useful? 

If you were selling a copy of the book used, which measure would you use in your ad? 

Simplify ratios when possible. 

• In the case that the denominator is 1, the ratio is called a unit rate. 

• Use Example 4 to show how a ratio can be more easily understood when simplified as a unit rate. 
The ratio is 100 miles to 3.2 gallons; or dividing by 3.2, a ratio of 31.25 miles per gallon. 

• Possible follow-up questions: Which ratio is easier to understand? Which is more convenient? 

• Point out that the interpretation of the ratio in Example 1, "The new book costs 20/13 times the cost 
of the used book", corresponds in fact to the unit rate ji/1 • 

Have students read the ratios aloud in Examples 1-4. For instance, in Example 2, ^ can be read as "4 to 
3" and can be written as 4 : 3,4/3, or g/1 

Compare and contrast Examples 2 and 3, paying close attention to the use of units. 

• In the first ratio, the units feet cancel; in the second, the units do not cancel. 

• It is worthwhile to point out that different units can be used in a ratio: miles per hour (mph), miles 
per gallon (mpg), and feet per second (ft/s) are common ratios. 

Use Example 6 to introduce proportions by setting the two ratios equal. The difference therefore between 
a ratio and a proportion is that ratios are expressions and proportions are equations. 

Students learn in Examples 7 and 8 that cross-multiplication is the correct way to solve an equation when 
the variable is in the denominator of a fraction. In an equation where the variable is in the numerator of 
the fraction such as y = I, it is not necessary to cross-multiply 

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Error Troubleshooting 

General Tip: Have students label the givens with proper units in all the proportions they setup, thereby 
avoiding the common mistake of solving a proportion with unlike units. 

• In Example 9, the length units and the time units must be the same. 

• The immediate next step for a student who sets up the proportion in Example 9 as 20 minutes = 7 hours 
is to convert the time units to a common unit. 

3.6 Scale and Indirect Measurement 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use scale on a map. 

• Solve problems using scale drawings. 

• Use similar figures to measure indirectly. 

Vocabulary 

Terms introduced in this lesson: 

indirect measurement 

scale 

scale drawings 

real distance 

scaled distance 

similar 

similar figures 

similar triangles 

Teaching Strategies and Tips 

This lesson introduces students to the basic mathematical methods used in map-making and scale drawings 
that rely on ratios and proportions. 

In Example 1, 

• The scale 1 cm = 1 km tells students that the distance in centimeters on the map is the same number 
of kilometers on the ground. 

• Point out that not all maps have such a simple scale. On those maps, proportions must be setup. 

m, 1 . <• 1 • 1 known distance on a diagram i ■ 1 j • 1 j r x, 

• the basic formula is: scale = — ■, -jt- : fn^, where the denominator is solved tor by cross- 

unknown distance in real lite ' J 

multiplying. 

In Example 2, have students use a calculator on the square root and remind them to round appropriately. 
Examples 3 and 4 use a different form of the basic formula: 

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• distance on a diagram = (distance in real life) x (scale) 

Example 5 requires use of a ruler. It is recommended that teachers prepare handouts or overheads in 
advance. See also Review Questions 1-3. 

Use Example 5 to point out that a standard centimeter rule will be divided into tenths whereas a U.S. 
inches rule will be divided into sixteenths. 

Use Examples 6 and 7 to introduce similar figures and indirect measurement. Possible motivating questions: 
How can the height of the Eiffel Tower or a lighthouse be determined? How are heights of trees or buildings 
measured? 

Error Troubleshooting 

General Tip: Students may have trouble setting up proportions in problems involving scaled drawings. 
The following may help: 

• The type of ruler (centimeter, inch) used is irrelevant. 

• If the scale is provided in the drawing (for example, 1 cm represents 183 m), then check that the 
drawing has not been enlarged or reduced from its original size by comparing the scale with an 
appropriate ruler. Then use 

distance on a diagram = (distance in real life) x (scale). 

• Otherwise, determine the scale by computing 

known distance on a diagram 



scale 



known distance in real life 



In Examples 6 and 7, corresponding pairs of triangle sides must correspond in the proportion. To help 
prevent students from setting up incorrect proportions, have them name the triangle sides: 

long side long side 
short side short side 

3.7 Percent Problems 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find a percent of a number. 

• Use the percent equation. 

• Find percent of change. 

Vocabulary 

Terms introduced in this lesson: 
percent 

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percent equation 

rate 

total 

part 

base unit 

positive percent change 

increase 

negative percent change 

decrease 

mark-up 

Teaching Strategies and Tips 

Work out exercises like Examples 1-7 and place the solutions in a table similar to the one below. This frees 
up board space and provides students with a handy reference to study later. 

Example. 

Complete the table. 

Table 3.1: 
Fraction Decimal Percent 

4 
5 

0.12 

2.5% 
0.01% 

3 

8 

0.001 

4 

>; 



Fraction, decimal, and percent conversions can be summed up as: 

• decimal => percent. Multiply by 100 and affix % symbol. Alternatively, move the decimal point two 
places right and affix % symbol. See Examples 1 and 2. Additional Example. 1.2 = : {qq = 120%. 

• percent => decimal. Divide by 100 and remove % symbol. Alternatively, move the decimal point two 
places left and remove % symbol. See Examples 3 and 4. Additional Example. 0.003% = ^yj|=p = 
0.00003. 

• fraction => percent. Represent the fraction as x% or x/100. Solve for x by cross-multiplying. Alter- 
natively, convert the fraction to a decimal. Then convert decimal to percent as above. See Examples 
5 and 6. Additional Example. § = ^ => x = 40. Therefore, § = 40%. 

• percent => fraction. Express percent as a ratio (per 100). Reduce fraction. See Example 7. Additional 
Example. 110% = ^ = n = i_i-. 

• fractions => decimals. Divide numerator by denominator. Use calculator if necessary. Example. 
.125 



■J 



1 

1.000 =>-= 0.125 
8 



decimals => fractions. Place the digits after the decimal under the appropriate power of ten and 

225 _ _9_ 
1000 40 • 



reduce. Example. 0.225 - ' ! v - - 



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In Examples 8-11, students setup percent equations to find the percent of a given number. 

• Convert the value for R% in the percent equation R% x Total = Part to a decimal before calculating. 
Rate should be expressed as a decimal in Rate x Total = Part. 

• Remind students that of means to multiply. 

Use Examples 12 and 13 to point out that a positive percent change means an increase in the quantity, and 
a negative change means a decrease. 

In Example 14, teachers are encouraged to work out the calculations for Mark-up, Final retail price, 20% 
discount, and 25% discount as a way to motivate the same calculations done algebraically in the next step. 

Additional Example. 

Is the order in which we calculate discounts and sales tax significant? In other words, should stores subtract 
the discount first and then add the tax on the new total or should the total amount be taxed first and then 
have the discount subtracted from that? or does it matter? Assume the discount is a percent and not a 
fixed discount. 

Solution. Consider an example: 

Original price of the item = $12.50 

Discount = 35% 

Sales tax rate in the county of purchase = 7.75% 

Tax First, Discount Second 

12.50 + 0.0775(12.50) = 13.47 
13.47-0.35(13.47) = 8.76 

Discount First, Tax Second 

12.50-0.35(12.50) = 8.13 
8.13 + 0.0775(8.13) = 8.76 

The steps above can be repeated algebraically. Let: 

p = original price of the item 

d = discount 

t = sales tax rate in the county of purchase 

Tax First, Discount Second 

p + t{p) = amount with tax 

p + t(p) - d{p + t{p)) = final amount with discount 

Discount First, Tax Second 

p — d{p) = amount after discount 

p - d{p) + t(p - d{p)) = final amount with discount 

Simplifying the two expressions for the final amount results in identical expressions (careful when distribut- 
ing). 

Conclusion: There is no difference in the total amount to be paid if the tax is added to the total first, 
followed by the discount, or the discount applied to the total first, followed by the tax. 

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General Tips: 

• Give students time to consider the possibilities on their own. 

• Have students choose their own numbers for original price, discount, and sales tax rate. This can be 
done in groups or individually. Calculators are recommended. 

• Asking around the classroom, it is suspicious that everyone's final two calculations are the same. Use 
this to formulate a conjecture. 

• Ask students to turn in their reasoning process as an assignment. 

• As the above algebraic argument is completely variable driven (no numbers), teachers are advised to 
show each step. 

• Students reason in various ways. Some common responses are: 

"Add the tax first, otherwise you are cheating the government out of its tax." 

"Taking the discount first decreases the bill, and so the tax will not be as great." 

"Tax should be added on last, as the discounted price is the true price of the item - since that is how much 
the item is being sold for." 

"Doing the tax first increases the amount to be paid and so the discount will be larger." According to 
students, this translates to a smaller price to be paid. 

"The quantities must be equal since they have seen it being done in both ways." 

As a class, discuss the validity of some of these responses. 

Error Troubleshooting 

Despite the % symbol in Example 4, students see the decimal and incorrectly move it two places right. 
This error is common for percents less than 1 (i.e., 0.5%) and percents greater than 100 (i.e., 110%). 

Additional Examples. 

Express 0.01% as a decimal. 

Solution: Move the decimal two places left and remove the % symbol. 0.01% = 0.0001. 

Express 120.25 as a percent. 

Solution: Move the decimal two places right and affix the % symbol. 120.25 = 1.2025%. 

3.8 Problem-Solving Strategies: Use a Formula 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Develop and apply the strategy: Use a Formula. 

• Plan and compare alternative approaches to solving problems. 

Vocabulary 

Terms introduced in this lesson: 

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procedure 

Teaching Strategies and Tips 

In this lesson, students solve applied problems using the four-step problem solving plan. 

In Example 2, providing a drawing of each stage of the climb may help students organize the given 
information. 

In Example 4, students can check their answers by working backwards. Consider the fraction remaining 
after each deduction. 

In the Review Questions, use the answer to Problem 6 to solve Problem 7. 

Error Troubleshooting 

General Tip: Remind students to interpret their answer in Step 4 of the problem solving plan. This gives 
students a moment to determine whether or not they've answered the question being asked in the problem. 

In Step 4 of Example 3, the discrepancy between the (approximate) solution and the actual time comes 
from rounding in Step 3. 



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Chapter 4 

TE Graphs of Equations and 
Functions 



Overview 

Formulas and equations describe the relationships between quantities. Graphs represent them visually. 
This chapter emphasizes reading and interpreting graphs. 

Suggested pacing: 

The Coordinate Plane - 1 hr 
Graphs of Linear Equations - 1 hr 
Graphing Using Intercepts - 1 - 2 hrs 
Slope and Rate of Change - 1 - 2 hrs 
Graphs Using Slope-Intercept Form - 1/2 hr 
Direct Variation Models - 1 hr 
Linear Function Graphs - 1/2 hr 
Problem-Solving Strategies: 
Use a Graph - 2 hrs 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

In this chapter, the problem-solving strategy is to Use a Graph. The examples and review questions 
encourage students to think about the data presented, to discuss reasonable tolerances for estimates, and 
to interpret graphs in real-life contexts. 

Alignment with the NCTM Process Standards 

The NCTM Process Standards in the use of graphs include portions of the communication, connections, 
and representation standards. One of the key requirements when constructing graphs is to organize and 
consolidate mathematical thinking in order to display the information accurately (COM.l). Graphing 
requires recognizing and using connections among mathematical ideas (CON.l) such as independent and 

www.ckl2.org 44 



dependent variables, appropriate scale when assigning values to the x— and v-axes, or patterns and trends 
in the data displayed. 

• C0M.1 - Organize and consolidate their mathematical thinking through communication. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• CON.l - Recognize and use connections among mathematical ideas. 

• CON. 3 - Recognize and apply mathematics in contexts outside of mathematics. 

• R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 



4.1 The Coordinate Plane 

Learning Objectives 

At the end of this lesson, students will be able to: 



Identify coordinates of points. 
Plot points in a coordinate plane. 
Graph a function given a table. 
Graph a function given a rule. 



Vocabulary 

Terms introduced in this lesson: 



coordinate plane 

x— and y-axes 

origin 

quadrant 

ordered pair 

x and y coordinates 

positive x, negative x 

positive y, negative y 

relation 

domain 

range 

graph of a function 

continuous function 

discrete function 

independent variable 

dependent variable 

linear relationship 

discrete problem 

slope 

intercept 



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Teaching Strategies and Tips 

Introduction: Motivate xy-coordinates with examples from daily life that employ rectangular coordinate 
systems. 

• Examples: a city map, the game of Battleship, a chessboard, spreadsheets, assigned seating at a 
theater. 

• Discuss how to find a particular location in each example: a seat in a theater can be found by row 
number and then by seat number. 

• Point out that the examples are lattices, differing from the Cartesian coordinate system in that they 
are discrete. 

Use Examples 1-3 to demonstrate finding coordinates of points on a graph and Examples 4 and 5 to plot 
points given their coordinates. Allow the class to make observations such as: 

• The coordinates of a point cannot be interchanged since the first coordinate specifies going left /right 
and the second coordinate, up/down. For example, (2,7) is not the same point as (7,2). 

• If a coordinate of a point is 0, then the point resides on an axis. 

• Quadrants can be distinguished by the signs of the coordinates contained in them. For example, a 
point having coordinates with the signs (— ,+) resides in quadrant II. Points with coordinates having 
signs (-, -) belong to quadrant III. 

• In Example 4, it is necessary to display four quadrants so that all points will be visible. The set of 
points in Example 5 have only positive coordinates; it is convenient therefore to display only the first 
quadrant. As an informal rule, axes do not need to be extended farther than the largest and smallest 
^-coordinates and y-coordinates. 

• Resize a graph by rescaling the axes. In general, the x and y-axes can be scaled differently. Axis 
tick marks do not need to be unit increments. 

General graphing tips: 

• In applied problems, the independent and dependent variables should be distinguished early. Ask: 
What quantity is depending on the other? 

• In setting up the axes, a suitable scale must be chosen. Ask: 

Will the important features of the graph be visible? 

Will it be necessary to use different increments along the two axes? 

• Constructing tables is a valuable tool. See Examples 6 &; 7. Allow students to use the simple inputs, 
x = 0, 1, -1, 2, in their tables when appropriate. 

The second method in Example 7 will be returned to in greater detail in a subsequent chapter. 

Error Troubleshooting 

General Tip: To determine the graph of a linear relationship, no more than two points are needed. Students 
can be encouraged to plot at least three to ensure no arithmetical errors were made. 

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4.2 Graphs of Linear Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph a linear function using an equation. 

• Write equations for, and graph, horizontal and vertical lines. 

• Analyze graphs of linear functions and read conversion graphs. 

Vocabulary 

Terms introduced in this lesson: 

formula 
equation 

Teaching Strategies and Tips 

In Example 1: 

• The coefficient of x represents the taxi's rate or cost per mile. Ask students to consider how the 
graph would change if the taxi charged more per mile, then less per mile. 

• The constant in the equation is the taxi's base fee. Ask students to interpret the y-intercept (the 
taxi's base fee). 

• Include units on the axes labels; i.e., x (in miles), y (in dollars). 

• Do the same for Example 2. Ask students to also interpret the x-intercept (number of years when 
the debt will be fully paid). 

In Examples 1 and 2: 

• Students make estimates and predictions from their graphs. Answers will be approximate, even 
though the equations can be solved exactly. Emphasis should be placed on reading and interpreting 
the graphs and not solving equations. 

• Students make graphs by constructing tables from the given equations. Have them plot at least 3 
points for accuracy. 

• Teachers may find it useful to show how the equations derive from the given information. 

Use Examples 3 and 4 to motivate the equations of horizontal and vertical lines. 

• The equations x = constant and y = constant are the simplest equations possible. 

• They are two- variable equations despite only one variable being present. The coordinate plane must 
be used to graph them. Often, students will see an equation like x = 3 and plot a single point on the 
x-axis at (3,0). 

• The equation y = 1, for example, signifies that the y quantity is fixed at 1 and x is free to take on 
any value. 

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• Given a horizontal line, the equation is v = the v-value of any point on the line. Analogously, vertical 
lines have the equation x = the x— value of any point on the line. This can help with Example 5. 

Additional Examples: 

What is the equation of the vertical line passing through (4, 5) ? 

Solution: x = 4. 

Given that a line passes through (2,-1) and is parallel to the x—axis, what is its equation? 

Solution: y = -1. 

Error Troubleshooting 

Allow students to distinguish between the input and output variables in Examples 6 and 7 before estimating. 
This way, they will know whether to begin correctly along the horizontal axis or the vertical axis. 

In the Review Questions, it may help students to rewrite y = 6 — 1.25x in Problem lc as y = -1.25x + 6. 

4.3 Graphing Using Intercepts 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find intercepts of the graph of an equation. 

• Use intercepts to graph an equation. 

• Solve real-world problems using intercepts of a graph. 

Vocabulary 

Terms introduced in this lesson: 

intercepts 

x-intercept, y-intercept 

standard form 

coefficients 

initial cost 

Teaching Strategies and Tips 

One way to graph lines is by plotting intercepts. Additional questions to ask students in the introduction: 

• Do all lines have x and y- intercepts? 
Solution: No. 

• Which lines have only one intercept? 
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Solution: Horizontal and vertical lines. 

• Is it possible for lines not to have intercepts? 
Solution: No, every line must have at least one intercept. 

• Is it possible for lines to have more than 2 intercepts? 

Solution: Yes, but infinitely many intercepts. The lines x = and y = cross the axes infinitely may 
times. 

In Examples 2a and 2b, have students convert their fractions into decimals before plotting. 

Use the cover-up method as a quick way to find intercepts algebraically. 

In Examples 4 and 5, encourage students to interpret the meaning of the intercepts in context of the 
problem. 

Error Troubleshooting 

General Tip: Students will often attempt to find an x-intercept by setting x = 0; and a y-intercept by 
plugging in for y. The opposite is true. To find an intercept, set the other variable to 0. 

In Example 4, distinguish between the independent and dependent variables first. To find an appropriate 
scale for the axes, determine the domain of the independent variable next. 

In Example 5, although "without spending more than $30" implies an inequality (translates as < 30), solve 
the problem as if it were an equality ("spending exactly $30") and then shade the triangular region. For 
more on systems of inequalities, see chapter Solving Systems of Equations and Inequalities. 

4.4 Slope and Rate of Change 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find positive and negative slopes. 

• Recognize and find slopes for horizontal and vertical lines. 

• Understand rates of change. 

• Interpret graphs and compare rates of change. 

Vocabulary 

Terms introduced in this lesson: 

slope 

positive slope, negative slope 

climbing, descending 

lattice points 

change 

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rate of change 

per 

undefined slope, infinite slope 

interpret a graph 

velocity 

Teaching Strategies and Tips 

Use the introduction to motivate the concept of slope. Point out: 

• Just as two points determine a unique line, a point and a slope also determine exactly 1 line. 

• Viewing the slope as the ratio, ^, is useful. From one point on the line, knowing how to rise and 
run brings you to a second point. 

• The slope of a line is constant. That is, for any two points on the line, the ratio ^^ is the same. 

• If m = g, then rise = 2 and run = 3. Since * = 5gj g° m g down two units and then left 3 units will 
also be a point on the line. 

Use Example 1 to demonstrate rise-to-run triangles for lines. The triangles are most useful when con- 
structed on lattice points (all coordinates of the vertices are integers). This makes the slope calculation 
effortless. Observe that the hypotenuse runs along the line. 

Use Example 2 to derive a formula for slope. 

Emphasize that graphs are read from left to right. 

• Linear functions are increasing when their graphs slant up and to the right (y increases as x is 
increased). In this case, slope is positive since Ay and Ax are both positive (or both negative). 

• Linear functions are decreasing when their graphs slant down and to the right (y decreases as x is 
increased). In this case, slope is negative since either Ay or Ax is negative, but not both. 

Supplement Example 4 with a skiing analogy. 

• Horizontal lines have zero slope, or no slope. This corresponds to cross-country skiing. 

• Vertical lines have undefined slope. This corresponds to falling down a cliff (undefined skiing). 
Vertical lines have infinite slope 

Error Troubleshooting 

General Tip. A common mistake is to subtract the x and y-coordinates in different orders in the slope 
formula; i.e., m + prj - - To avoid making this error, students can write point 1 and point 2 above the two 
points, and then select the coordinates in the same order. See Example 6. Of course, the choice for point 
1 and point 2 is arbitrary. 

4.5 Graphs Using Slope-Intercept Form 

Learning Objectives 

At the end of this lesson, students will be able to: 
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• Identify the slope and y-intercept of equations and graphs. 

• Graph an equation in slope-intercept form. 

• Understand what happens when you change the slope or intercept of a line. 

• Identify parallel lines from their equations. 

Vocabulary 

Terms introduced in this lesson: 

slope-intercept form 

rise 

run 

parallel lines 

Teaching Strategies and Tips 

Use Examples 1 and 2 to make observations such as: 

• m < when a line slants downward and m > when it slants upward. 

• m = when a line is horizontal. 

• b < when the y-intercept is below the x-axis and b > when it's above the x-axis. 

• b = when a line passes through the origin. 

Use the slope-intercept method to graph lines as an alternative to plotting and joining two intercepts. 

With a graphing utility, demonstrate the effects on a line when changing m and b one at a time in an 
equation in slope-intercept form. Make observations such as: 

• The larger the m, the steeper the line. 

• Negative slopes can also represent steep lines. The smaller the m (more negative), the steeper the 
line. 

• Slopes approximately equal to zero represent lines that are almost horizontal. 

• Changing the intercept shifts a line up/down. 

• Parallel lines have the same slope but different y-intercepts. 

Error Troubleshooting 

In Example 2, use the marked lattice points and/or intercepts in the slope calculation for each line. Using 
these points allows students to obtain exact answers. See also Review Questions, Problems 2 and 3. 

4.6 Direct Variation Models 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify direct variation. 

• Graph direct variation equations. 

• Solve real-world problems using direct variation models. 

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Vocabulary 

Terms introduced in this lesson: 

direct variation 

constant of proportionality 

directly proportional 

Teaching Strategies and Tips 

Use Examples 1-3 to motivate direct variation. 

• Direct variation is an expression of a simple linear relationship with v-intercept = 0. Since the line 
passes through origin, (0,0) is a point on it. 

• The slope of the line is the constant of proportionality and is the only parameter in the problem. 
Therefore, it takes only one more point to determine the line (to determine the constant of propor- 
tionality) . 

• Emphasize the switch from m (slope) to k (constant of proportionality) is only notational. 

Determine the constant of proportionality by substituting the given information (look for one point) into 
y = kx. The result is the equation of variation; use it to answer the question in the problem. 

By solving applied problems involving direct variation, teachers demonstrate the usefulness of these models. 

Additional Examples: 

The cost of raisins varies directly with the number of kilograms of raisins purchased. If the cost is $34.25 
when the number of kilograms purchased is 7.5, calculate the amount of raisins that can be purchased for 
$10.50. 

Solution: Let C denote the cost of the raisins and N the number of kilograms of raisins purchased. The 
direct variation equation is C = k ■ N. Substitute C = 34.25, N = 7.5. 

34.25 = k ■ 7.5 
4.6 «Jt 

With the constant of proportionality known, calculate N when C = 10.50. 

C = A.Q-N 
10.5 = 4.6-JV 
2.2 = N 

So the number of kilograms of raisins that can be purchased for $10.50 is 2.2 kg. 

A ball was thrown from the roof of a high building. The velocity v of the falling ball is directly proportional 
to the time t of the fall. After 4 seconds, the velocity of the ball is 85 meters per second. What will the 
velocity be after 6 seconds, assuming it hasn't hit the ground yet? 

Hint: Solve for k in the direct variation equation v = k ■ t first. 

Error Troubleshooting 

NONE 

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4.7 Linear Function Graphs 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Recognize and use function notation. 

• Graph a linear function. 

• Change slope and intercepts of function graphs. 

• Analyze graphs of real- world functions. 

Vocabulary 

Terms introduced in this lesson: 

vertical line test 
slope-intercept form 
arithmetic progression 
common difference 
discrete 
continuous 

Teaching Strategies and Tips 

Students learn to use function notation for the first time. 

• f(x) is pronounced "/ of x". 

• y = f(x), u y is a function of x", and "y depends on x" are synonymous. Also, the input or independent 
variable is x. 

• "/(•*)" an d "y = " are interchangeable. This means that the graphing techniques students have 
learned previously can be used to graph functions. 

Point out in Example 1 that the y variable depends on x; that's the only reason for solving for y in each 
case. 

Additional Example: 

Rewrite the following equation using function notation if cost depends on the number of pounds purchased. 

C-3-2n = 

Solution: Solve for C and replace C with C(n). 

C = 2n + 3 

C(n) = 2n + 3 

Use Example 2 to show how function notation is used. There is nothing new computationally; students 
evaluate expressions as they before. Remind students to use order of operations. 

x-intercepts can be found by solving f{x) = and y-intercepts by computing /(0). See Example 4. 

Use Example 5 to show how linear functions and arithmetic sequences are related. 

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• In an arithmetic sequence, terms are found by adding the same constant to the previous term. 

• For linear functions, when the input variable is increased by 1, the output variable changes by the 
value of the slope. 

Use Example 5 to distinguish between discrete and continuous. Use examples from daily life: The number 
of pumpkin seeds in a pumpkin is discrete. The number of people that can fill a football stadium is discrete. 
The waiting time for a bus at a bus stop is continuous. The weights of newborn babies are continuous. 

Error Troubleshooting 

General Tip. f{x) is function notation and should not be confused with multiplication (/ times x is not 
correct). 

Example 5c and Problems 6b and 6c of the Review Questions have no consecutive terms to subtract to 
determine the common difference. In such an event, use unknowns for the sequence terms starting with 
the first given term. Solve the resulting equation. See the hint at the end of Example 5c. 

4.8 Problem-Solving Strategies - Graphs 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Use the strategy: Read a graph. 

• Develop and apply the strategy: Make a graph. 

• Solve real-world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 
approximate answer 

Teaching Strategies and Tips 

Remind the students of the four-step problem-solving plan: 

Step 1: Understand the Problem 

Read the problem carefully. Once the problem is read, list all the components and data that are involved. 
This is where you will be assigning your variables. 

Step 2: Devise a Plan - Translate 

Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart or construct 
a table as a start to solving your problem. 

Step 3: Carry Out the Plan - Solve 

This is where you solve the equation you came up with in Step 2. 

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Step 4: Look - Check and Interpret 

Check to see if your answer makes sense. 

Emphasize that Step 4 entails more than "going over your work" to see "if it's all there." Cursory checking 
often misses the error because the same mistake is made again. Above all, encourage students to determine 
whether their answer seems reasonable. 

Error Troubleshooting 

General Tip. Have students draw fairly precise graphs to obtain better approximate answers. Use rulers, 
sharp pencils, graphing paper, and consistent increments on the axes. 



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Chapter 5 

TE Writing Linear Equations 



Overview 

Students apply their knowledge about linear equations to solve real-world problems. They use linear 
regression methods to fit lines to the data provided and make predictions. 

Suggested pacing: 

Linear Equations in Slope- Intercept Form - 1 hr 
Linear Equations in Point-Slope Form - 1 hr 
Linear Equations in Standard Form - 1 - 2 hrs 
Equations of Parallel and Perpendicular Lines - 1 - 2 hrs 
Fitting a Line to Data - 0.5 hrs 
Predicting with Linear Models - 1 hr 
Problem Solving Strategies: 
Use a Linear Model - 2 hrs 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

In this chapter, the problem solving technique Use a Linear Model builds directly on lesson material, 
particularly in "Fitting a Line to Data" and "Predicting with Linear Models." Within the context of this 
chapter, linear modeling is defined as using linear interpolation, linear extrapolation, or a line of best fit 
as a method of predicting trends and/or obtaining reasonable data. 

Alignment with the NCTM Process Standards 

Being able to approximate or estimate well (R.2) is a valuable skill in mathematics as well as real life. When 
younger students are asked to estimate, they often follow the rules for rounding rather than truly estimating 
with regard to the magnitude of a quantity (CON.l). Sometimes teachers unintentionally contribute to this 
issue because it is difficult to correct estimations; several estimations could be acceptable given different 
scenarios or different priorities. Taking a few moments to discuss significant digits in real- life situations, 

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such as the cost of a house, a car, or a meal at a restaurant, can really improve students' number sense 
and their ability to make appropriate approximations (COM. 2; COM. 3). 

Informal scale drawings can be very helpful whenever geometric shapes are part of a class exercise, and, 
when done attentively, can internalize the notion of scale (R.l). Free-hand enlargements or miniatures, 
which many students love to do, can develop an instinct for proportional reasoning (RP.4) and engage 
artistically inclined students. Displaying attractive, correct student work around the room reinforces the 
concept of scale and inspires others to think proportionately (R.3). 

The question of the reasonableness of a solution is something that must be addressed repeatedly. Teachers 
should ask students to reflect on the reasonableness of their answers on a regular basis (PS. 4). 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• CON.l - Recognize and use connections among mathematical ideas. 

• PS. 4 - Monitor and reflect on the process of mathematical problem solving. 

• RP.4 - Select and use various types of reasoning and methods of proof. 

• R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

• R.2 - Select, apply, and translate among mathematical representations to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

5.1 Linear Equations in Slope-Intercept Form 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write an equation given slope and y-intercept. 

• Write an equation given the slope and a point. 

• Write an equation given two points. 

• Write a linear function in slope-intercept form. 

• Solve real-world problems using linear models in slope-intercept form. 

Vocabulary 

Terms introduced in this lesson: 
slope-intercept form 

Teaching Strategies and Tips 

Slope-intercept form: 

• Allows quick identification of the slope and y-intercept. This makes graphing linear functions easier, 
for example. 

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• Written so that the dependent variable is isolated. This makes finding ordered pairs easier, for 
example. 

Students learn to write linear equations in slope-intercept form, given: 

• The slope and y-intercept of the line. See Examples 1,2, and 7. 

• The slope and any one point on the line {b is not given). See Examples 3 and 8. 

• Any two points on the line (neither m nor b are given). See Examples 4 and 9. 

In Example 2, have students check their answer by choosing two different points on each line and finding 
the slope. Construct triangles using lattice points. 

In Example 4, suggest that students label their ordered pairs, writing X\,y\ and X2,y2 above each coordinate. 

Use Example 5 to demonstrate plugging an expression into a function. Encourage students to keep the 
parentheses with the expression as it gets plugged into the function. 

Additional Example: 

Let f(x) = -2x + 3. Find /(-4jc + 1). 

Solution: The entire expression (-4x + 1) must be plugged into the function. Keep the parentheses as you 
plug in: 

/((-4x + l)) = -2(-4x + l) + 3 
/((-4x + 1)) = 8x - 2 + 3 = 8x + 1 

Check for student conceptual understanding of function notation. For example, 

• /(— 2) = 5 is equivalent to the ordered pair (-2, 5) or x = -2 and y = 5. 

• /(0) = 5 is a y-intercept. /(3) = is an x-intercept. 

Assume that Examples 7-9 can be modeled by a linear function. 

• The slope and y-intercept have contextual significance in applied problems. Have students interpret 
each in context of the problem. 

• In applied problems, students can single out the slope from among the givens by looking for key 
phrases that signify a slope. For example, dollars per hour (Example 7) and inches per day (Example 

8). 

Error Troubleshooting 

In Example 2, remind students that a run left or a rise down are negative quantities. Students commit 
the error in part a. for example when they go down 1, right 1 but write the slope as +1 instead of — 1. 

In Example 3b, remind students to find common denominators. 

In Example 4b, remind students to watch the double negative in the denominator: —^ 



2-(-4) • 

5.2 Linear Equations in Point-Slope Form 

Learning Objectives 

At the end of this lesson, students will be able to: 
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• Write an equation in point-slope form. 

• Graph an equation in point-slope form. 

• Write a linear function in point-slope form. 

• Solve real-world problems using linear models in point-slope form. 

Vocabulary 

Terms introduced in this lesson: 
point-slope form 

Teaching Strategies and Tips 

Students learn to write linear equations in point- slope form given: 

• The slope and any one point on the line (possibly the y-intercept). See Examples 1, 2, and 8. 

• Any two points on the line (m is not given). See Examples 3 and 7. 

An equation in point-slope form: 

• Uses subscripts on x and y to designate the fixed, given point, x and y assume any other points on 
the line. 

• Is not solved for y. Suggest that students generate other values of y by solving for y first. 

• Can be used to graph the line without having to rewrite the equation in slope-intercept form because 
a slope and a point determine a unique line. See Example 5. 

Use Example 3 to show that any point on the line can be substituted for (xo,yo). Point-slope equations 
will simplify to the same slope-intercept equation regardless of the chosen point. 

Use Example 6 to introduce function notation for equations in point-slope form. 

• Remind students that /(5.5) = 12.5 is equivalent to the ordered pair (5.5,12.5) in 6b. 

"Flat fees", initial amounts, starting times, etc. correspond to the intercept along the vertical axis. 

Error Troubleshooting 

In Example 7, have students determine the independent and dependent variables first. This helps them 
form correct ordered pairs. 

5.3 Linear Equations in Standard Form 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write equivalent equations in standard form. 

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• Find the slope and y-intercept from an equation in standard form. 

• Write equations in standard form from a graph. 

• Solve real-world problems using linear models in standard form. 

Vocabulary 

Terms introduced in this lesson: 

standard form 

slope 

intercepts 

Teaching Strategies and Tips 

An equation in standard form: 

• Can be used to express the equation of a vertical line. This is not possible in slope-intercept or 
point-slope forms. 

• Makes finding intercepts easy. Remind students of the cover-up method introduced in lesson Graphing 
Using Intercepts, chapter Graphs of Equations and Functions. 

Use Example 3 to find the equations of the lines in standard form using the intercepts and without resorting 
to slope. The steps are essentially the steps used in the cover-up method only in reverse. 

Have students derive the equations that describe the situations in Examples 4 and 5. 

Error Troubleshooting 

General Tip: Point out that the b in ax + by = c does not represent the y-intercept as it did in an equation 
in slope-intercept form. 

In Example 3c, the ^-intercept is a fraction, x = |. After eliminating the denominator, 2x = 3, proceed 
with the method as usual using the 3 instead; i.e. What number times 3 equals some other number times 
4 (the y-intercept)? 

General Tip: Encourage students to include units when labeling their variables. 

5.4 Equations of Parallel and Perpendicular Lines 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Determine whether lines are parallel or perpendicular. 

• Write equations of perpendicular lines. 

• Write equations of parallel lines. 

• Investigate families of lines. 

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Vocabulary 

Terms introduced in this lesson: 

parallel lines 
perpendicular lines 
family of lines 
vertical shift 

Teaching Strategies and Tips 

Use the introduction to make observations such as: 

• Parallel lines have different y-intercepts. 

• Parallel lines have the same slope. Allow students to come to this conclusion using a rise-over-run 
argument: If one line runs (or rises) more than another line, then the lines cannot be parallel; they 
will eventually meet. 

• Perpendicular lines have opposite reciprocal slopes. Encourage students to draw two perpendicular 
lines. Since one line will be increasing and the other decreasing, this shows that the slopes have 
opposite signs. Have students also construct the rise-over-run triangles for each line. This will 
demonstrate that the rise of one line is the run of the other and vice-versa; the slopes are also 
reciprocal. 

Have students recognize in Example 5 that y = -2 is the equation of a horizontal line. The problem can 
then be recast as finding the equation of a vertical line through (4, -2). 

Additional Examples: 

What is the equation of a line parallel to the x—axis and passing through (2,-1) ? 

Solution: A line parallel to the x-axis is horizontal and therefore has equation, y = k. Since it passes 
through (2,-l),y = — 1. 

Find the equation of a line perpendicular to x = 3. 

Solution: Since x = 3 is a vertical line, a perpendicular line will be horizontal. Therefore any equation of 
the form y = k is perpendicular to x = 3. 

Find the equation of a line perpendicular to x = 3 that passes through the point (14, 15). 

Solution: y = 15. 

Use Example 9 and a graphing utility to investigate families of lines. 

• Fixing the slope m and varying the y-intercept b in y = mx + b results in a set of parallel lines; one 
through each point on the y-axis. 

• Fixing the y-intercept and varying the slope results in all possible lines through that y-intercept; 
one for every real number angle. 

Error Troubleshooting 

NONE 

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5.5 Fitting a Line to Data 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Make a scatterplot. 

• Fit a line to data and write an equation for that line. 

• Perform linear regression with a graphing calculator. 

• Solve real-world problems using linear models of scattered data. 

Vocabulary 

Terms introduced in this lesson: 

scatterplot 

measurement error 

outlier 

linear regression, line of best fit 

Teaching Strategies and Tips 

Before fitting a line to data, suggest that students always check the scatterplot first for signs of association 
between the variables. If the points are too scattered, any calculations will be meaningless. 

Use the introduction and Example 1 to motivate linear regression. 

• Data points are collected from measurement or experiment which is never perfect. 

• A line of best fit is the line closest to all the data points. This means that the line minimizes the 
square root of the sum of the squares of the distances from each point to the line. 

Provide the class with examples they can relate to. For example: 

• Compile a list of students' ages and heights. 

• Use a graphing utility to make a scatterplot of age vs. height. 

• Ask the class whether the data set looks approximately linear. If so, perform a linear regression using 
a graphing utility. 

• Use the line of best fit to make predictions of height from ages. Have students gather similar age and 
height data from their older siblings as a way to check the model. 

Error Troubleshooting 

In Examples 3 and 4, discourage students from using any two data points to determine the equation of the 
line of best fit. 

• Two points on the line or two data points closest to the line must be used. 

• Teachers are encouraged to show that the line of best fit does not always pass through every point 
in the data set; in fact, it is possible that the line doesn't go through any data point. 

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5.6 Predicting with Linear Models 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Collect and organize data. 

• Interpolate using an equation. 

• Extrapolate using an equation. 

• Predict using an equation. 

Vocabulary 

Terms introduced in this lesson: 

surveys 

experimental measurements 

non-linear data 

linear interpolation 

polynomial interpolation 

linear extrapolation 

most accurate method 

Teaching Strategies and Tips 

Use the introduction to motivate data collection and organization. 

• Data are gathered from surveys and experimental measurements. 

• Data are organized via tables and scatterplots, where it is easier to spot trends and patterns. 

In Example 1: 

• Point out that two data sets are being displayed simultaneously in the scatterplot. This is a common 
practice when two data sets are being compared. 

• The two variables are Median Age of Males and Females At First Marriage by Year. 

• Discuss with students whether the scatterplot is approximately linear and whether using a line of 
best fit to predict future values is appropriate. Do the same for Example 2. 

Use Examples 3 and 4 to motivate linear interpolation. 

• Ask students how they would go about estimating a value where there is no data point available. 

• Possible discussion questions: Assume the data are linear. How would the line of best fit help? Should 
only a subset of the data be used? All of the data? How does the above considerations change for 
non-linear data? 

Use Example 5 to motivate linear extrapolation. 

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• Point out that the last data point is an outlier and therefore influences the extrapolation heavily. 

• Work through the extrapolation a second time using a linear regression. Have students compare 
answers from the two models. 

• Emphasize that extrapolation is not useful when used to predict values far into the future (or far 
into the past). 

For additional data sets, visit: 

• http://www.census.gov 

• http://www.cdc.gov 

Error Troubleshooting 

General Tip: On the TI graphing calculators, students should be using LinReg(ax + b) or LinReg{a + bx) 
to perform linear regressions and not LnReg. 

General Tip: Students can neglect to consider the accuracy of a prediction or estimate. Some estimates 
will not be good because the desired value is far removed from the rest of the data set. 

5.7 Problem Solving Strategies: Use a Linear Model 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Develop and apply the strategy: Use a Linear Model. 

• Plan and compare alternative approaches to solving problems. 

• Solve real-world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 
linear modeling 

Teaching Strategies and Tips 

In Example 1: 

• The problem combines a question from geometry with a solution from algebra. 

• Assume a linear association exists between circumference and diameter, which is plausible given the 
scatterplot. 

• Allow students to pick their own points to find the line of best fit. Compare regression models as a 
class. 

• As a class, compute one interpolation and one extrapolation. Compare with the results obtained 
from the line of best fit. 

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• All models will contain some error due to measurement. Convince students that the v-intercept 
should be in any model: a circle of zero diameter has no circumference. 

Observe in the scatterplot in Example 2 that the data are not linear. 

• Assume that the data do not change drastically between the known data points. 

• Interpolation or extrapolation methods will give better estimates than the line of best fit. 

Error Troubleshooting 

General Tip: Before performing a linear regression on a data set, have students always check the scatterplot 
first for approximately linear data. 



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Chapter 6 

TE Graphing Linear 
Inequalities 

Overview 

In this chapter, students solve linear inequalities. They use the addition and multiplication properties of 
inequality and learn that the direction of the inequality is sometimes reversed when solving inequalities. 
Students solve single-step and multi-step inequalities and progress to compound inequalities and absolute 
value equations and inequalities. The chapter finishes with graphing inequalities in two variables. 

Suggesting Pacing: 

Inequalities Using Addition and Subtraction 1 hr 

Inequalities Using Multiplication and Division - 1 hr 

Multi-Step Inequalities - 1 - 2 hrs 

Compound Inequalities - 1 - 2 hrs 

Absolute Value Equations - 0.5 hrs 

Absolute Value Inequalities - 1 hr 

Linear Inequalities in Two Variables - 0.5 hr 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

Teachers can increase students' problem solving abilities by presenting them with challenging activities. 

A Sudoku warm-up, used on a daily basis for a short period of time or once a week, can lead to excellent 
conversations on strategies. Solving problems together, students share their thinking and learn from each 
other. Furthermore, students can continue with Sudoku outside of class and practice various approaches. 

Teachers can find Tiling Checkerboard Challenges or other thought-provoking brainteasers on the web and 
can easily differentiate them to meet the needs of students. Students can make progress with Checkerboard 
Tiling problems in many different ways. When asked, "Can an ordinary 8x8 checkerboard be covered by 
31 dominoes if two squares are removed?" students can sketch solutions, use paper tiles, use graph paper, 
or, more abstractly, simply consider the two-color arrangements on the checkerboard. There's an entry 
point for every student, regardless of his or her mathematical background. 

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Alignment with the NCTM Process Standards 

Playing mathematically oriented games such as Sudoku, chess, Hex, Mudcracky, Five-In-a-Row, and Set 
can incorporate many of the NCTM process standards, particularly the Communication Standard. In a 
structured classroom setting, students will organize and consolidate their thinking (COM.l), communicate 
coherently and clearly to peers, teachers, and others (COM. 2), and analyze and evaluate the thinking and 
strategies of others (COM. 3). In addition, they will use representations to model and interpret physical 
phenomena (R.3). Both within and outside the classroom, students will apply and adapt a variety of 
appropriate strategies to solve problems (PS. 3) and select and use various types of reasoning and methods 
of proof (RP.4). 

• COM.l - Organize and consolidate their mathematical thinking through communication. 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• RP.4 - Select and use various types of reasoning and methods of proof. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

6.1 Inequalities Using Addition and Subtraction 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write and graph inequalities with one variable on a number line. 

• Solve an inequality using addition. 

• Solve an inequality using subtraction. 

Vocabulary 

Terms introduced in this lesson: 

inequality 

interval, interval of values 

Teaching Strategies and Tips 

Use Examples 1 and 2 to introduce the number line as a way to graph the solution set to an inequality. 

• Point out that the solution sets in Examples 1 and 2 represent all numbers which make the statements 
true. The solution to a linear equation is a number; the solution to an inequality, an interval of 
(infinite) numbers. 

• Remind students that an open circle is used for inequalities containing a > or < symbol and a closed 
circle for inequalities containing a > or < symbol. 

Additional Examples: 

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Graph the following inequalities on the number line. 

a. x < -6 

b. x>3 

c. x > 1 

d. x < 10 

Write the inequality that is represented by each graph. 



a. *-*- 



-10 -5 5 10 



b. «- 



-1000 -800 -600 -400 -200 200 400 600 800 1000 



C. «-t- 



-10 -8-6-4-2 2 4 6 8 10 

d. «— i — ■ — i — ■ — i 1 1 1 1 1 ( — ■ — i — - — i-* 

-100 -80 -60 -40 -20 20 40 60 80 100 

In Example 3, students learn to identify inequalities in sentences. The following chart might be useful for 
those students having difficulty choosing the correct symbol. 



< 


> 


< 


> 


less than 


greater than 


at most 


at least 


fewer 


more than 


no more than 


no less than 






less than or equal to 


greater than or equal to 



Additional Examples: 

Write each statement as an inequality and graph it on the number line. 

• You were told not to spend any more than $20 at the arcade. 

• Fewer than 200 tickets are available for sale to the musical performance. 

• You must be taller than 40 inches to get on this ride. 

• The FDA allows for 30 or more insect fragments per 100 grams of peanut butter. 

Use Examples 4 and 5 to show students how to isolate variables in inequalities using addition and subtrac- 
tion. 

• Point out that solving inequalities is analogous to solving equations. 

• The exception occurs when multiplying or dividing by a negative number. See the lesson Inequalities 
Using Multiplication and Division. 



Error Troubleshooting 



General Tip: Suggest to students having difficulty with inequalities that the inequality opens to the larger 
number. 

• a < b is read as "a is less than b" or u b is greater than a," depending on the perspective. 

• In a statement such as 2 < x, suggest that students take the point of view of the unknown: "jc is 
greater than 2" or "all real numbers greater than 2" instead of "2 is less than x". 

In Example 5d, remind students of mixed number form and that -| - 5 = — 5|. 

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6.2 Inequalities Using Multiplication and Divi- 
sion 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve an inequality using multiplication. 

• Solve an inequality using division. 

• Multiply or divide an inequality by a negative number. 

Vocabulary 

Terms introduced in this lesson: 

inequality notation 

set notation 

interval notation 

solution graph 

open and closed brackets 

including/not including 

reversing the inequality 

Teaching Strategies and Tips 

Use a simple example to present the four ways of expressing the solution set of an inequality: 
Solve for x: 

3x < -24 
%x -24 

x< -8 

Solution: 

• Inequality notation: x < -8. 

• Set notation: The answer is {x\x a real number, x < -8}. 

• Interval notation: (-00,-8]. A combination of parentheses and brackets are used. Because numbers 
less than -8 are solutions, negative infinity is used. 

• Solution graph: 



-10 -so 5 10 

The answer is expressed as a closed circle (solid dot) at -8 and shaded to the left. 
Tips on set notation: 



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The vertical bar separates the variable from the condition that is used to describe the set. It is read 

as "such that." 

{x\x a real number, x < -8} is read as "the set of all values of x, such that x is a real number less 

than or equal to 2." 

Show students that set notation takes the general form: { "variable" such that "condition is true" }. 

Students may not see the value of using set notation for the set {x\x a real number, x < -8} when 

inequality notation suffices. Suggest that although shading a number line or expressing an answer in 

interval notation may be easier in this example, set notation has advantages in other examples. 



Discrete sets are easily described using set notation: 

The number of pets belonging to students in class {0, 1,2,3,4} 

The solutions to an equation {-1,2}. 

The set of prime numbers {2, 3, 5, 7, 11, 13, 17, . . .}. 

Tips on interval notation: 



This is the only form of solution that uses the infinity symbol. 

Point out that infinity is always paired with parentheses and never a bracket. The reason for this 

is because infinity is not a number and only a concept. (There is no end to the number line, so it 

cannot be included in the solution set.) 

The two extreme cases of interval notation are (a, a) which represents the single number a, and 

(-oo, oo) which represents all real numbers. 



Use Example 2 to show that inequality signs change direction when multiplying or dividing by negative 
numbers. 



Explain why the rule is necessary. 



1. The number line is constructed so that the negative side is a reflection of the positive side. Multiplying 
or dividing a number by a negative is equivalent to reflecting it across the origin, as through a mirror. 

2. For any two positive real numbers on the number line, one will be further to the right than the other. 
Multiplying or dividing them by a negative has the effect of reflecting them across the origin. The 
rightmost number on the positive side of the origin becomes the leftmost number on the negative 
side of the origin. 

3. Therefore, if a is greater than b, then -a is less than —b. This means that the inequality sign changes 
direction when multiplying or dividing by a negative. 

4. In the case that one number is negative and the other positive, a simpler argument holds. 



Error Troubleshooting 

General Tip. Remind students to reverse the inequality sign when multiplying or dividing an inequality 
by a negative number. 

Because the right side of the inequality in Example 2b has a fraction, suggest that students multiply both 
sides by -1/9 to avoid dividing the fraction. 

Additional Example: 

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2 

-6x< - 

9 

(-- J (-6x) < I -- J I - J The direction of the inequality is changed. 

1 
x > 

27 

In Examples 2b and 2d, remind students that the direction of the inequality does not change on account 
of the negative sign on the right side of the inequality. For example, when solving for x in 12x > -30, do 
not change the direction of the inequality. 

6.3 Multi-Step Inequalities 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a two-step inequality. 

• Solve a multi-step inequality. 

• Identify the number of solutions of an inequality. 

• Solve real- world problems using inequalities. 

Vocabulary 

Terms introduced in this lesson: 

two-step inequality 
multi-step inequality 
multiple solutions 
no solutions 
discrete solutions 

Teaching Strategies and Tips 

In solving multi-step inequalities, remind students to follow the order of operations in each step. 

In general, the steps used to solve an inequality are the same as the steps used to solve an equation. The 
one exception is reversing the inequality sign when multiplying or dividing by a negative. 

Use Example 4 to show that an inequality can have a finite and discrete solution set. Compare and contrast 
this with previous inequalities having an infinite solution set. 

Inequalities can have various types of solutions: 

• The solution set of 2x > 10 is the infinite set [5,oo). 

• The solution set of 12 + x < x + 12 is the infinite set (-oo, -oo) (all real numbers). 

• In the next lesson, students learn that the infinite set [-1, 7) is a solution set to an inequality. 

• The inequality 12 - x < —x + 3 has no solutions. 

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• Inequalities that model real-world problems in which the variables represent integer quantities (usu- 
ally positive), have discrete solution sets. In Example 4, the solution set is {0,1,2,3,4}, a finite, 
discrete set. In Examples 5 and 6, the solution sets are infinite, discrete sets: 

{35, 36, 37, . . .} and {146, 147, 148, . . .}, respectively. 

Error Troubleshooting 

In Example 3a and Problems 6-10 in the Review Questions, remind students to follow the order of opera- 
tions. Clear parentheses first. 

When multiplying both sides of the inequality by 4 in Example 3b, remind students to multiply both terms 
on the right by 4. 

In Example 6, 

• Remind students that the given numbers must be in the same unit, dollars (or cents). 

• Remind students to round their answers up to the next highest integer. Ask them why this is 
necessary. 

Additional Example: 

A local bicycle shop advertises bikes for as low as $235. Ted decides to save his lunch money to purchase 
one. If he puts away 85 cents daily (including weekends), in how many days will he be able to bring home 
a bike? 

Hint: Convert 85 cents to 0.85 dollars. Round the answer up to 277 days. 

6.4 Compound Inequalities 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write and graph compound inequalities on a number line. 

• Solve a compound inequality with "and." 

• Solve a compound inequality with "or." 

• Solve compound inequalities using a graphing calculator (TI family). 

• Solve real-world problems using compound inequalities. 

Vocabulary 

Terms introduced in this lesson: 

compound inequalities 
combined inequalities 

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Teaching Strategies and Tips 

Use this lesson to show how two or more inequalities can be combined with "and" or "or." 
As a class, discuss the differences between "and" and "or." 

• Use real- world examples: 

Earth has one moon and Venus has no moons. (True) 
Earth has two moons and Venus has no moons. (False) 
Earth has two moons or Venus has no moons. (True) 
Earth has two moons or Venus has three moons. (False) 

• Allow the discussion to lead to when "and" statements are true and when "or" statements are true. 
Introduce compound inequalities in a way similar to single inequalities. 

• Have students rewrite compound inequalities as individual inequalities and then solve each one sep- 
arately. 

• Solutions to "and" are solutions satisfying both inequalities. Solutions to "or" are solutions satisfying 
at least one inequality. See Example 1. 

• Suggest that students graph each part of a compound inequality on two separate parallel number 
lines, aligned at the origin. The solution set can be graphed on a third. 

Example: 

Graph the following compound inequality on the number line. 

x < -4 or x > 1. 

Solution: 

Begin by drawing three parallel number lines. On the first, graph the solution set to the first part, x < -4. 
On the second, graph the solution set to the second part, x > 1. Since the statement is joined with an "or", 
solutions to the compound inequality, x < — 4 or x > 1, are those in at least one of the two solutions sets. 



-• — £ — t — * — « — * — ■ — • — * — • — •- 
-5-4-3-2-1012345 



-• — • — * — *- 



-5-4-3-2-10 12 3 4 5 



* — * — £ — • — # — * — * — 0-^— —• ■ ■ ■ ■ 

-5-4-3-2-10 12 3 4 5 

As a way to check an answer, have students choose one point from each shaded interval. Test these points 
in the main inequality. Also, test the endpoints to make sure the correct dot was used (solid or hollow). 

Use Examples la, Id, 2a, and 2d to introduce compact notation for "and." For example, x < 4 and x > -4 
is rewritten as -4 < x < 4. 

Allow students in Examples lb, lc, 2b, and 2c to infer that a solution set consisting of two branches is 
"or." 

Compound inequalities with "and" can be solved without breaking them into two individual inequalities. 
See Example 3. 

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Example: 

Solve the compound inequality. 



-1 <2x-3<7. 



Solution: Without rewriting the compound inequality as two separate inequalities, solve for x by adding 3 
to each of the three "sides". 

-l + 3<2x-3 + 3<7 + 3 
2 < 2x < 10 

Divide each of the three "sides" by 2. 

2 %x 10 

2 % 2 
1 <x< 5. 

As students have trouble with "and" and "or," encourage them to think of "and" as what's common to 
two solution sets. (Ask: What's common to this interval and that interval?) Have them think of "or" as 
the combination of two intervals. (Ask: What do you get when you put both intervals together?) 

Check for conceptual understanding with exercises such as: 

Example: 

Graph the following compound inequalities on the number line. 

a. x < -3 and x > 2 

b. x < -3 and x < 2 

c. x > -3 and x > 2 

d. x > -3 and x < 2 

e. x < -3 or x > 2 

f. x < -3 or x < 2 

g. x > -3 or x > 2 

h. x > -3 or x < 2 

The exercises above exhaust all the possible cases using "and", "or", " < ", " > " and two arbitrary numbers. 
Students see the effects of varying one characteristic at a time. 

Error Troubleshooting 

General Tip. Remind students to use the correct symbols: 

Table 6.1: 



Interval notation Inequality notation Solution graph 



Included endpoint [,] >,< solid dots 

Excluded endpoint (, ) >, < hollow dots 



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General Tip. Encourage proper notation. 

• An answer such as -1 < x > -5 is either incomplete (not simplified) or incorrect. It is unclear whether 
the student intended on using "and" or "or". 

• In Examples 6 and 7, discourage students from writing 1.875 > t > 1.56 and 8.25 < t < 6.75 . Remind 
students to reverse the entire expression in their final answer to 1.56 < t < 1.875 and 6.75 < t < 8.25. 

6.5 Absolute Value Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve an absolute value equation. 

• Analyze solutions to absolute value equations. 

• Graph absolute value functions. 

• Solve real- world problems using absolute value equations. 

Vocabulary 

Terms introduced in this lesson: 

absolute value 

absolute value equations 

vertex or cusp 

Teaching Strategies and Tips 

Focus on the interpretation of absolute value as a distance. 

• In Example lb, | - 120| = 120 because -120 is 120 units from the origin. 

• In Example 2, have students explain, using a distance argument, why the order in which the two 
numbers are subtracted is not important. In general, for any two numbers (or points) a and b, 
\a- b\ = \b - a\. 

• Have students rethink the simple absolute value equations \x\ = 3 and |x| = 10 in Example 3 as |?| = 3 
and |?| = 10, respectively. (Which numbers are 3 units from the origin? 10 units from the origin?) 

• Have students interpret absolute value equations out loud. \x — 2| = 7 means "those numbers on the 
number line 7 units away from 2." See Examples 4-6. Encourage students to draw the number line 
and mark the possible solutions. 

• Using the distance interpretation, point out that absolute value equations (involving only linear 
functions) can have no more than 2 solutions. Have students consider absolute value equations with 
1 or solutions such as \x - 2| = and \x - 2| = 5, respectively. 

Students have trouble reconciling the definition "|x| = —x if x is negative" and the fact that "absolute value 
changes a negative number into its positive inverse." Offer an example: 

• Let x = -5. Then | -5| = -(-5) since -5 is negative (using the definition). This simplifies to | -5| = 5. 

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• I - 5| = 5 since absolute value changes a negative number into its positive inverse. 

Use Examples 5 and 6 to show students how to rewrite absolute value equations so that the distance 
interpretation is clearer. 

Additional Examples: 

a. Solve the equation and interpret the answer. 

|x+l| = 3 

Solution: As it stands, the equation cannot be interpreted in terms of distance. Rewrite the equation 
with a minus sign: \x - (-1)1 = 3 which can now be interpreted as those numbers 3 units away from -1. 
Therefore, the solution set is {2, -4}. 

b. Solve the equation and interpret the answer. 

|3x-6| = 8 

Hint: As it stands, the equation cannot be interpreted in terms of distance. Rewrite the equation by 
dividing both sides by 3: 

U_ 6 _ 8 
T ~ 3 ~ 3 

be -21 = - 
3 

This last equation can now be interpreted as those numbers 8/3 units away from 2. 
Treat the absolute value as a grouping symbol when appropriate. 



h. _ 6 

f, 3 



The distributive law holds in expressions such as 3|x — 4| = |3x — 12| and o|3x - 6| 

The distributive law does not hold in an expression such as — 2\x - 3| + \ — 2x + 6| 

In general, distribute into absolute value a • \b\ = \a ■ b\ when a = \a\; i.e., for positive numbers a. 

These steps are based on the property \a\ ■ \b\ = \a ■ b\. 



x-2\. 



When beginning to graph absolute value functions, encourage students to make a table of values such as 
those in Examples 7 and 8. 

Have students plot and describe in words the basic graph v = \x\. Allow them to observe the essential 
properties of the absolute value graph: 

• The graph has a "V" shape, consisting of two rays that meet at a sharp point, called the vertex or 
cusp. 

• One side of the "V" has positive slope and other side negative slope. 

• The vertex is located at the point where the expression inside the absolute value is equal to zero. 

Error Troubleshooting 

General Tip: Remind students not to distribute a negative into the absolute value expression. For example, 
-2|x-3| *|-2x + 6|. 

General Tip: Students may misinterpret "absolute value is always positive" and commit the error — 13 — 5| = 
2. Suggest to students that in such situations -|3-5| = (-1)|3-5|. The multiplication by -1 happens after 
the absolute value has been performed, and so -|3 - 5| = (-1)|3 - 5| = (-1)1 - 2| = (-1)2 = -2. 

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6.6 Absolute Value Inequalities 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve absolute value inequalities. 

• Rewrite and solve absolute value inequalities as compound inequalities. 

• Solve real-world problems using absolute value inequalities. 

Vocabulary 

Terms introduced in this lesson: 
absolute value inequality 

Teaching Strategies and Tips 

Use Example 1 to show that the distance interpretation equally applies to absolute value inequalities. 
Additional Examples: 

a. Solve the inequality. 

\x\ < 10. 

Solution: |x| < 10 represents all numbers whose distance from the origin is less than or equal to 10. This 
means that -10 < x < 10. 

b. Solve the inequality. 

|x| > 10. 

Solution: |x| > 10 represents all numbers whose distance from the origin is greater than or equal to 10. This 
means that x < -10 or x > 10. 

Use Example 1 and the distance interpretation to motivate solving the absolute value inequalities in 
Examples 2-5. 

• Allow students to infer from Example 1 that |x| < a <=> —a < x < a and |x| > a <=> x < —a or x > a. 
In Problems 4 and 5 in the Review Questions, have students divide by the coefficient of x first. 

Error Troubleshooting 

NONE 

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6.7 Linear Inequalities in Two Variables 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph linear inequalities in one variable on the coordinate plane. 

• Graph linear inequalities in two variables. 

• Solve real-world problems using linear inequalities. 

Vocabulary 

Terms introduced in this lesson: 
dashed line/solid line 

Teaching Strategies and Tips 

In this lesson, students learn to graph linear inequalities on the coordinate plane. 

• Students draw dashed lines for the strict inequalities (<, >); interpret this as excluding the points on 
the line from the solution set. Students draw solid lines for the inequalities <,>; interpret this as 
including the points on the line in the solution set. 

• Have students shade those regions of the plane which satisfy the given inequalities. 

• As a general rule, shade above the line y = mx + b if the stated inequality is y > mx + b. Shade below 
the line if y < mx + b. Have students solve for y first. See Examples 5-7. 

• Graphing inequalities on the coordinate plane is a step up from graphing on the number line and 
requires more care. 

Use Examples 1 and 2 to motivate graphing the absolute value inequalities in Examples 3 and 4. 

• Have students rewrite the absolute value inequality as a compound inequality first. 
In Examples 8 and 9 and Problems 13 and 14 in the Review Questions, 

• Point out that quadrant I is the only quadrant used because the variables should be positive. 

• Only the points with integer coordinates are possible solutions in Example 9. 

Error Troubleshooting 

General Tip. Students can misinterpret an inequality such as x > 2 as an inequality in one variable and 
incorrectly shade that part of the x-axis for which x > 2. 



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Chapter 7 

TE Solving Systems of 
Equations and Inequalities 

Overview 

In this chapter, students discover the methods that determine solutions to systems of linear equations and 
inequalities. Students begin by solving systems of equations graphically, realizing that the solution for 
each system is the point of intersection between the two lines represented by the given equations. 

Suggested Pacing: 

Linear Systems by Graphing - 1 hr 

Solving Linear Systems by Substitution - 1 hr 

Solving Linear Systems by Elimination through Addition or Subtraction - 1 hr 

Solving Systems of Equations by Multiplication - 1 - 2 hrs 

Special Types of Linear Systems - 0.5 hr 

Systems of Linear Inequalities - 2 hrs 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

In this chapter, Systems of Equations and Inequalities, several different methods of solving equations 
and inequalities are taught. Throughout, real-world problems are incorporated into the exercises, and 
practical applications of important concepts such as constraints, optimum solutions, feasibility regions, 
and maximum and minimum values are outlined. 

In this chapter you might try giving students limited choices about which problems to do independently. 
This allows you to assess which of the problems students feel comfortable with, which they wish to avoid, 
and where some re-teaching might be helpful. With the section Comparing Methods of Solving Linear 
Systems, students might be asked to use all three primary methods on a single exercise rather than doing 
three different exercises with the same method. In addition to giving necessary practice it may help 
students become more discerning about which method is simpler in a given situation. 

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Alignment with the NCTM Process Standards 

Using the principle of choice and, occasionally, asking students to write about a problem of their preference 
can address many of the NCTM Process Standards. Chief among them would be the Communications and 
Connections standards. Students solving and presenting solutions for equation and inequality problems 
must organize and consolidate their mathematical thinking (COM.l), communicate their mathematical 
thinking coherently and clearly to peers, teachers, and others (COM. 2), and use the language of mathe- 
matics to express mathematical ideas precisely (COM. 4). When students share their various insights and 
approaches to problems as common classroom practice, they learn to analyze and evaluate the mathemat- 
ical thinking and strategies of others (COM. 3). 

When students are allowed to select specific problems as part of an assignment rather than to merely mimic 
the sample problems given in the text, they strive to recognize and use connections among mathematical 
ideas (CON.l) and to understand how mathematical ideas interconnect and build upon one another to 
produce a coherent whole (CON. 2). They are encouraged to apply and adapt a variety of appropriate 
strategies to solve problems (PS. 3) and use representations to model and interpret physical, social, and 
mathematical phenomena (R.3). 

• COM.l - Organize and consolidate their mathematical thinking through communication. 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• COM. 4 - Use the language of mathematics to express mathematical ideas precisely. 

• CON.l - Recognize and use connections among mathematical ideas. 

• CON. 2 - Understand how mathematical ideas interconnect and build on one another to produce a 
coherent whole. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

7.1 Linear Systems by Graphing 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Determine whether an ordered pair is a solution to a system of equations. 

• Solve a system of equations graphically. 

• Solve a system of equations graphically with a graphing calculator. 

• Solve word problems using systems of equations. 

Vocabulary 

Terms introduced in this lesson: 

system of equations 
solution to an equation 
solution to a system of equations 
point of intersection 

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Teaching Strategies and Tips 

Present students with a basic problem to motivate systems of equations. 

Example: 

Find two numbers, x and y, such that their sum is 10 and their difference is 4. 

Allow students some time to find the numbers. Encourage guess- an d-check at first. A good place to start 
is with pairs of integers. 

Solution: 

The problem can be translated as: 

x + y = 10 
x - y = 4 

Ask: Of all the possible ordered pair solutions to the first equation, which also satisfy the second? 

x y sum difference 

5 5 10 

6 4 10 2 

7 3 10 4V 

Additional Example: 

Complete the table for each equation. Compare the rows of the two tables to determine the solution to the 
system. 

2x + y = 8 
x -y = 1 

x y 



1 

2 

3 

x y 


1 
2 
3 

Hint: Solve each equation for y first. 

Use the introduction and Example 1 to point out that a system of equations is one problem despite there 
being two equations. 

• The two equations must be solved "together" or simultaneously. 

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• The problem is not done until both x and y have been determined. 

• The solution to an equation is a number; the solution to a system of equations is an ordered pair. 

• An ordered pair solution satisfies, or "makes the equations true." 

Additional Examples: 

Find the solution to the following systems of equations by checking each of the choices in the list. 

x + y = 3 
2x + y = 1 

i. (2,1) 
ii. (5,-2) 
hi. (-2,5)V 
b. 



7x + y = 7 
-3x - 2y = -14 

i. (1,0) 

h. (0,7)V 

hi. (4,3) 

Use Examples 2-4 to demonstrate the graphing method for solving a system of equations. 

• Lines can be graphed using any method: constructing a table of values, graphing equations in slope- 
intercept form, solving for and plotting the intercepts. 

Emphasize that the graphing method approximates solutions. 

• It is exact when the point of intersection has integer coordinates or easily discernible rational numbers. 

• Suggest that students draw careful graphs. 

• By zooming in, a calculator provides the coordinates of the intersection point to any degree of 
accuracy although the solution can still be approximate. 

Error Troubleshooting 

General Tip: To generate y values, as for a table, have students solve each equation for y hrst. See Example 
6. 

General Tip: To demonstrate that an ordered pair is a solution to a system, remind students that it must 
satisfy both equations. To demonstrate that an ordered pair is not a solution to a system, remind students 
that at least one of the equations will not be satisfied. 

7.2 Solving Linear Systems by Substitution 

Learning Objectives 

At the end of this lesson, students will be able to: 
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• Solve systems of equations in two variables by substituting for either variable. 

• Manipulate standard form equations to isolate a single variable. 

• Solve real- world problems using systems of equations. 

• Solve mixture problems using systems of equations. 

Vocabulary 

Terms introduced in this lesson: 

substitution 

substitution method 

standard form of a linear equation 

Teaching Strategies and Tips 

Use Example 2 to motivate the substitution method. 

• As the solution consists of fractions, the system is more complicated than any example presented up 
to this point. 

• Emphasize that the graphing method can only provide an approximation. Therefore a different 
method is needed. 

The substitution method: 

• An algebraic method; provides exact solutions. 

• A technique for replacing an unknown with another expression to obtain a third equation with only 
one unknown (replacing equals with equals). 

• Best used when one of the coefficients of the variables is 1. 

Encourage students to isolate the variable with a coefficient of 1 or — 1. Students often give themselves 
extra work by choosing an equation and a variable at random. 

Mixture problems: 

• Mixtures do not necessarily pertain to chemistry. See Example 4 and Review Question 7. 

• Approach Example 7 with a picture. By the labeling the unknowns in it, the system of equations 
will be evident. 

Error Troubleshooting 

In Example 2, have students back-substitute x into one of the original equations in case that an error was 
made in solving for x. 

General Tip: In fact, any of the two original equations can be used to find y once x is determined. The 
easier- looking the equation the better. 

Point out in Example 3 that the question can be answered after determining x = 117.65. There is no need 
to back-substitute to find the cost per month. 

General Tip: Remind students to write coordinates in correct order, depending on how they labeled the 
variables in the beginning. 

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• Students often incorrectly write the value they found first as x. 

• For problems where variables other than x and y are used, have students clearly state which variable 
is independent and which is dependent. See Review Questions 5-9. 

7.3 Solving Linear Systems by Elimination through 
Addition or Subtraction 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a linear system of equations using elimination by addition. 

• Solve a linear system of equations using elimination by subtraction. 

• Solve real- world problems using linear systems by elimination. 

Vocabulary 

Terms introduced in this lesson: 

elimination 

method of elimination 

coefficient 

Teaching Strategies and Tips 

Use Example 1 to motivate the elimination method. 

• To find the cost of one banana, it makes sense to subtract the equations. 

• In general, equations can be added or subtracted so that a variable cancels. 

Show that in Example 2, equations can be added column- wise; each column representing a different variable. 
When x or v cancel, use Ox or Oy as placeholders. 

In Example 3, 

• Drawing a picture helps. 

• Encourage students to label variables first. The two unknowns are the speed of the river and the 
speed of the canoe in still water. 

• Compare going downstream to walking on a moving walkway in an airport terminal. Travelers walk 
faster than usual as their speeds are boosted by the walkway. The opposite is true going upstream 
and against the moving walkway. 

Error Troubleshooting 

Remind students in Review Problem 3 to align the variables column-wise. 
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In Review Problems 4, 6, 7, and 9, encourage students to put the equation being subtracted in parentheses. 
This way, it will be easier to remember to distribute the negative. 

General Tip: When subtracting two equations, remind students not to forget to subtract the constants on 
the right side. 

7.4 Solving Systems of Equations by Multiplica- 
tion 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a linear system by multiplying one equation. 

• Solve a linear system of equations by multiplying both equations. 

• Compare methods for solving linear systems. 

• Solve real- world problems using linear systems by any method. 

Vocabulary 

Terms introduced in this lesson: 

scalar 

lowest common multiple 

Teaching Strategies and Tips 

Use the introduction to convince students that the elimination method applies to any linear system because 
one or both equations can be multiplied by a constant resulting in a "new" pair of equations with matching 
coefficients. 

In Example 1, some students have trouble keeping track of the multipliers for each equation. Try using a 
visual: 

7x + Ay = 17 same 7x + 4y = 17 

5x-2y = ll ^ 10x-4y = 22 

In Example 2, 

• Point out that the distance covered is the same in both directions, so the 400 yards is unnecessary 
information. 

• Remind students to back-substitute to complete the problem. 

Use Example 3 to demonstrate a system with no matching coefficients and no coefficients that are multiples 
of others. 

• Remind students how to find the LCM of two numbers. 

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• Suggest that students find the LCM of the "smaiier pair" of coefficients, 3 and 5, instead of 880 and 
1845. 

Teachers may decide to forgo back-substitution and instead teach elimination of the second variable ("dou- 
ble elimination"). 

Additional Example: 

Solve the system using multiplication. 

2x - 5y = 8 
-llx+3y = 15 



Solution by "double elimination". Eliminate x first. 



Add. 



2x - 5y = 8 ^ 22x - 55y = 88 

-llx + 3y = 15 ^ -22x + 6y = 30 



22x - 55y = 88 
- 22x + 6y =30 

0x-49y =118 



Divide by -49. Therefore, y = —-r§ 



To find x, eliminate y next. 



2x - 5y = 8 ^ 6x - 15v = 24 

-llx + 3y=15 ^ -55x+15y = 75 



Add. 



6x - 15v = 24 

- 55x + 15y = 75 

- 49x + Oy =99 

Divide by -49. Therefore, x = -||. 

Answer: The solution to the system is ( - fi> - ^r)- 

The advantage of "double elimination" is that the fraction does not need to be back-substituted. 

Error Troubleshooting 

General Tip: Students forget to multiply every term in an equation by the scalar. 

General Tip: Encourage students to eliminate the variable whose coefficients in both equations have the 
smallest LCM. 

Remind students in Review Problem 2e to align the variables column- wise. 
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7.5 Special Types of Linear Systems 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify and understand what is meant by an inconsistent linear system. 

• Identify and understand what is meant by a consistent linear system. 

• Identify and understand what is meant by a dependent linear system. 

Vocabulary 

Terms introduced in this lesson: 

consistent system 

inconsistent system 

infinite number of solutions 

dependent system 

determining the system type (graphically, algebraically) 

Teaching Strategies and Tips 

Use this lesson to classify systems of equations according to the number of solutions they have. 

• Encourage students to use the graphical interpretation and rewrite equations in slope-intercept form 
first to compare slopes. See Examples 1-4. 

Use Examples 5 and 6 to show how to classify a system using the substitution or elimination methods. 
Solving inconsistent or dependent systems by substitution or elimination leads to variables dropping-out. 

• A false statement, such as 3 = 4, indicates an inconsistent system (parallel lines). 

• A true statement, such as -12 = -12, indicates dependent system (coinciding lines). 

When students arrive at an answer such as 8 = 8, they may assign 8 to one of the variables and solve for 
the other variable. 

Use Example 6 to point out that the equations in a dependent system are multiples of each other. 

Additional Example: 

Solve the system by multiplication. 

3x = —y - 5 

2y + 10 = -6x 



Solution: Align the variables column-wise. 



3x + y = -5 
Qx + 2y = -10 



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Eliminate y by multiplying the first equation by -2. 



Add. 



x(-2) 

3x + y = -5 > -6x - 2y = 10 

6x + 2y = -10 same 6x + 2y = -10 



- 6x - 2y = 10 
6x + 2y = -10 

Ox + Oy =0 



.»2 



Therefore, the system is dependent. Notice that the equations are multiples of one other: 3x + y = -5 
6x+2y= -10. 

In Examples 7-9, emphasize the geometric interpretation in context: 

• In Example 7, the lines intersect because the rental fees are different for the two membership options. 

• In Example 8, the lines are parallel because the memberships have different flat fees (y-intercepts) , 
but the same rental fee (slope). 

• In Example 9, the lines coincide because the equations use the same information. It is not possible 
to determine the price of each fruit since the second equation does not give any new information. 

Error Troubleshooting 

General Tip: In dependent systems, students sometimes misinterpret "an infinite number of solutions" to 
mean that any ordered pair will satisfy the given system of equations. Of course, only the infinite number 
of points on the line that the equations represent are solutions. 

7.6 Systems of Linear Inequalities 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph linear inequalities in two variables. 

• Solve systems of linear inequalities. 

• Solve optimization problems. 

Vocabulary 

Terms introduced in this lesson: 

system of inequalities 

half-plane 

dotted line/ solid line 

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bounded solution/ unbounded solution 
linear programming 

constraints 
feasibility region 

optimization equation 
maximum/minimum value 

Teaching Strategies and Tips 

Have students follow Example 1 step-by-step for the first few Review Questions. 

• Shade each region differently. 

Encourage students to rewrite each equation in slope-intercept form. This will help them graph the line 
and decide which half-plane to shade. 

Use Example 2 as an illustration of a system of inequalities with no solution. 

• Because the lines are parallel, the shaded regions will never intersect. 

• It is possible, however, for lines to be parallel and have shaded regions intersect. For instance, reverse 
the inequalities in Example 2. 

In Example 3, 

• Emphasize that the method used to determine solutions to a system of inequalities can be extended 
to any number of inequalities. 

• Point out that the pair of inequalities, x > and v > describes the first quadrant of the coordinate 
plane. In fact, any quadrant can be similarly described: 

Quadrant I: x > and v > 

Quadrant II: x < and v > 

Quadrant III: x < and v < 

Quadrant IV: x > and y < 

Have students follow Examples 5 and 6 step-by-step for Review Question 8. 

Error Troubleshooting 

General Tip: Remind students to reverse the direction of the inequality sign when multiplying or dividing 
by a negative number. See Review Questions 1-7. 



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Chapter 8 

TE Exponential Functions 



Overview 

Students learn to simplify expressions involving exponents. They work with scientific notation and then 
move on to exponential growth and decay and finally geometric sequences. 

Suggested Pacing: 

Exponent Properties Involving Products - 1 hr 

Exponent Properties Involving Quotients - 1 hr 

Zero, Negative, and Fractional Exponents - 0.5 hr 

Scientific Notation - 1 hr 

Exponential Growth Functions - 1 hr 

Exponential Decay Functions - 1 hr 

Geometric Sequences and Exponential Functions - 1 - 2 hrs 

Problem-Solving Strategies: 

reprise Make a Table; Look for a Pattern - 2 hrs 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

This chapter, Exponential Functions, begins with the essentials of exponential notation and builds sequen- 
tially through fairly sophisticated uses of exponents to solve real-world problems. The patterns related 
to exponential notation provide an opportunity to demonstrate the power of definitions. Looking for a 
pattern and using a table for projecting compound interest, for example, allows the repeated factor to be 
discovered and affirms that an exponent is used appropriately in the formula. 

Alignment with the NCTM Process Standards 

NCTM Process Standards from every strand can be seen in the lesson Problem-Solving Strategies. Students 
will both analyze and evaluate their own mathematical thinking and the strategies of others (COM. 3) and 
use the language of mathematics to express mathematical ideas precisely (COM. 4). In tackling business and 
scientific real world problems, they will make connections, recognize and apply mathematics in contexts 

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outside of mathematics (CON. 3). From a problem-solving perspective students will apply and adapt a 
variety of appropriate strategies to solve problems that arise in mathematics and in other contexts (PS. 2, 
PS. 3), and they will recognize reasoning and proof — specifically in applying the use of defining terms — as 
fundamental aspects of mathematics (RP.l). In developing tables to help them formulate their conclusions, 
students will create and use representations to organize, record and communicate mathematical ideas (R.l) 
and use these representations to model and interpret physical, social, and mathematical phenomena (R.3). 
This is a rich problem-solving lesson indeed! 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• COM. 4 - Use the language of mathematics to express mathematical ideas precisely. 

• CON. 3 - Recognize and apply mathematics in contexts outside of mathematics. 

• PS. 2 - Solve problems that arise in mathematics and in other contexts. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• RP.l - Recognize reasoning and proof as fundamental aspects of mathematics. 

• R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

8.1 Exponent Properties Involving Products 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use the product of a power property. 

• Use the power of a product property. 

• Simplify expressions involving product properties of exponents. 

Vocabulary 

Terms introduced in this lesson: 

power 

exponent 

square, cube 

base 

factors of the base 

product rule for exponents 

power of a product 

power rule for exponents 

Teaching Strategies and Tips 

Encourage students to review basic exponents in the chapter Real Numbers. 

• An exponent is a notation for repeated multiplication of a number, variable, or expression. 

• Exponents count how many bases there are in a product 

• Parentheses precede exponents in the order of operations. 

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Additional Examples: 
Write in exponent form. 

a. (4)(4)(4) 

b. (-7) (-7) (-7) (-7) 

c. (5w)(5w)(5w) 

d. t-t-t-t-t-t 

Encourage proper use of mathematical language: 

• 3 4 is read as "three to the fourth power," "three to the fourth," or "three raised to the power of four." 
The exponents 2 and 3 are special: 3 2 , "three squared" and 3 3 , "three cubed." 

• (3x) 3 is read as "quantity 3x cubed." 

• -3 2 is read as "opposite of three squared." 

• (-3) 2 is read as "negative three squared". 

To check for conceptual understanding, ask students to translate the following into symbols. 

• Square negative two. Answer: (-2) 2 

• Negative two squared. Answer: (-2) 2 

• The opposite of two squared. Answer: -2 2 

Allow the class to infer the product rule for exponents in Example 2. 

• "When you multiply like bases you add the exponents." 
Allow the class to infer the power rule for exponents in Example 6c. 

• "When you raise an exponent to an exponent, you multiply them." 
Combine exponent rules in Examples 6-9. 

• Suggest that students apply each rule one step at a time. 

• Order of operations must be followed at each step: evaluate exponents before multiplying. See 
Examples 6a, 6b, and 9. 

Additional Examples: 

Simplify the following expressions. 

a. (4x)(3x) 2 

Hint: Apply the exponent before multiplying. 

b. (5x 2 + (-2x) 2 ) 3 

Hint: Simplify in the parentheses first. 

c. (4x)(3x) 2 + (5x 2 + (-2x) 2 ) 3 
Hint: Simplify each term first. 

Point out in Example 8 where the commutative property of multiplication is being used. 

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Error Troubleshooting 

Use Examples 4 and 5 to point out two common errors: 

• Multiplying bases incorrectly (exponents are different). 2 4 • 2 3 + 4 7 . 
However, 3 4 • 2 4 = 6 4 because 3 4 • 2 4 = (3 • 2) 4 = 6 4 (exponents are same). 

• Applying the product rule incorrectly (bases are different). 3 4 • 2 3 £ 6 7 . 

Remind students in Review Question 6 that even powers of negative numbers are positive. Try using a 
visual to show that negatives cancel in pairs: 

(_2)6 = (_ 2 )(-2)(-2)(-2)(-2)(-2) = (-2)(-2) ■ (-2)(-2) ■ (-2)(-2) = 64 

+4 +4 +4 

General Tip: Occasionally, students forget that: 

• negative X positive = negative 

• negative X negative = positive 

In Review Question 13, remind students that the exponent in -2y 4 does not apply to -2. Therefore, —1y 
is simplified. 

Additional Examples: 

a. Explain the difference between Ax 2 and (4x) 2 . 

b. Explain the difference between -l 2 and (-1) 2 . 

Teachers are encouraged to survey the class for answers. Ask: What constitutes the base of an exponent? 
General Tip: Encourage students to think about syntax before inputting an expression into a calculator. 

• (-1) 2 often gets inputted incorrectly as -1 A 2. 

• (oj can get inputted incorrectly as 2/3 A 2. 

8.2 Exponent Properties Involving Quotients 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use the quotient of powers property. 

• Use the power of a quotient property. 

• Simplify expressions involving quotient properties of exponents. 

Vocabulary 

Terms introduced in this lesson: 

quotient rule for exponents 
power rule for quotients 

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Teaching Strategies and Tips 

Allow students to infer in Example 1 that canceling like factors is equivalent to finding the difference in 
the exponents of the factors. 

• Therefore, the quotient rule for exponents is based on canceling like factors. 

Additional Example: 

Simplify the expression using the quotient rule. 

t. 

x 3 ' 

Solution: 

X 3 / ■ / ■ / 1 

Notice also that: 

K = x 4-3 = x 1 = x by the quotient rule. 

Use Example 2 to show what happens when a larger exponent is subtracted from a smaller exponent. 

• Encourage students to do the division longhand. 

• Allow students to come to the conclusion that positive exponents result when subtracting smaller 
exponents from larger exponents. 

Additional Example: 

Simplify the expression leaving all powers positive. 



m 3 n 2 
nfin 3 



Solution: Subtract the smaller exponents from the larger exponents and leave the result in the denominator 
(where the larger exponents were). 



m 3 n 2 



Use Examples lc, 3b, and 6 to show that the quotient rule is applied separately for each factor in the 
expression. 

Additional Examples: 
a 2xy2 

6x 3 y 
b 72x 5 y 2 z 



2 z 4 

2„2-5 



56.ryr 
C - 18* 8 .yz 

Combine the quotient and power rules in Examples 5-7. 

• Suggest that students include each step in their solution. 
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• Remind students to follow the order of operations. 

Additional Examples: 

a f24m 3 f (V 2 \ 2 
d - V 10m 4 ) \ rv> J 

b fl2^f.2i 

t 65x*f f 
c - \U5x?y 2 ) 

Error Troubleshooting 

In Review Question 7, have students apply the quotient rule first. This will save students some work in 
simplifying because the numbers will be smaller. See also Examples 4a and 4b; and Review Questions 11, 
13, and 14. 

8.3 Zero, Negative, and Fractional Exponents 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Simplify expressions with zero exponents. 

• Simplify expressions with negative exponents. 

• Simplify expression with fractional exponents. 

• Evaluate exponential expressions. 

Vocabulary 

Terms introduced in this lesson: 

zero rule for exponents 

negative exponents 

negative power rule for exponents 

fractional exponents 

radical/ root 

Teaching Strategies and Tips 

Use this lesson to show how exponent rules apply to zero, negative, rational, and irrational powers. 

Use Examples 1 and 2 to show that moving a factor over a fraction bar changes the sign of its exponent. 
Draw the analogy to moving a term over the equal sign. 

Example: 

x xy~ 2 

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Remind students that negative exponents turn an expression into its reciprocal with a positive exponent 
and have nothing to do with the overall expression's sign. 

Example: 

(5x)- 3 + -J^s, but (5x)- 3 - - 1 - '- 



(5x)3> """ V"-V ~ (5.v) 3 ~~ 125*3- 
Some students may try to incorrectly cancel the two negatives in an expression such as (-2)~ 5 . Point out 
that (-2)- 5 t {y2)/ 5 , but (-2)~ 5 = ^ = --L. 

Additional Examples: 

Write the following expressions without fractions. 



a. 


xy 


b. 




c. 


12m 
»2 



Write the following expressions without negative exponents. 

a. ti — rr 

b. -*&- 



12y" 5 

c. 8- 2 a- 2 ^ 3 c- 4 



Write the following expressions without negative exponents. If an expression cannot be simplified, write 
"cannot be simplified". 



a. 


x k 


b. 


x- k 


c. 


-x k 


d. 


(-*)* 


c. 


-x- k 



f. (-x)- k 

In Example 3, suggest that students apply the exponent rules on each number and variable separately, 
which is possible by the commutative property. See also Review Questions 10-16. 

Point out that there are two ways to begin Review Question 9. 

• For problems that can be simplified in several ways, it may not always be clear to students which 
step should be first. 

Work out additional examples: 

Simplify the following expressions so that no negative exponents appear in the answer. 

a. (x _2 y 3 ) -5 

Hint: Either use the power rule first or take care of the negative exponents first. 

Hint: Either simplify inside parentheses first or use the power rule. 

«■ m 2 

Hint: Simplifying in the parentheses first will save a few steps. 
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d. (xh 6 ) 

Hint: Apply the power rule first, thereby avoiding having to make common denominators later. 

e. (Optional) ((x 2 )" 5 ) 3 

Hint: Either multiply the two inner exponents first or multiply the two outer exponents first. 

Encourage students to ask why zero, negative, rational, and irrational powers are all valid exponents. If 
the following definitions are made: x° = l,x~" = ^, and x» = yx™ then the product, quotient, and power 
rules hold for all real numbers. 

Error Troubleshooting 

In Example 5a, remind students to make common denominators. 

General Tip: In more complicated expressions as in Example 7, have students write out each step line-by- 
line to avoid making algebraic errors. 

General Tip: Remind students to write a 1 (not 0) after completely canceling a factor. For example, 
i = -L 

8.4 Scientific Notation 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write numbers in scientific notation. 

• Evaluate expressions in scientific notation. 

• Evaluate expressions in scientific notation using a graphing calculator. 

Vocabulary 

Terms introduced in this lesson: 

power of 10 

unit 

scientific notation 

decimal point 

Teaching Strategies and Tips 

The basic idea in this lesson shows students that moving a number's decimal point is equivalent to multi- 
plying by a power of 10. 

• The advantage is to shorten long numbers. 

• Long numbers can be large or small. 

Additional Examples: 

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Write the numbers in scientific notation. 

a. 314159000 

Solution: Move the decimal point left 8 places until there is only one digit to the left of it. Compensate 
for moving the decimal left (same as dividing) by multiplying by an equivalent number of 10s. Answer: 
3.14159 x10 s . 

b. 0.00000234 

Solution: Move the decimal point right 6 places until there is only one digit to the left of it. Compensate 
for moving the decimal right (same as multiplying) by dividing by an equivalent number of 10s. Answer: 
2.34 xl0~ 6 . 

There are two steps for writing a number in scientific notation: 

• Step 1: Count the number of places the decimal point is moved until there is one (non-zero) digit to 
the left of the new decimal point. 

• Step 2: Multiply the new number by 10 to the power of the count in step 1. Make this exponent 
positive if the decimal point moved left, and negative if moved right. 

Remind students that an integer such as 34, 542, 100 has an implied decimal point after the last digit 0. 

Have students read out loud numbers in scientific notation. For example, read 7.45 x 10 8 as "7.45 times 10 
to the power of 8". 

Error Troubleshooting 

Remind students in Example 7 that 1% must be written as j™. 

General Tip. Remind students that the power of 10 does not apply to the number in front. For example, 
6.3 x 10~ 4 * g^YoT, but 6.3 x 10~ 4 = f$. 

8.5 Exponential Growth Functions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph an exponential growth function. 

• Compare graphs of exponential growth functions. 

• Solve real-world problems involving exponential growth. 

Vocabulary 

Terms introduced in this lesson: 

exponential function 
exponential growth 
exponentially increasing 

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Teaching Strategies and Tips 

To help students see the differences between different functions, graph y = 2 X , y = x, and y = x 2 side- by-side. 
Use a table of values for each function. 

Students may not make the connection between v = 2 X and a quantity that doubles. Have students 
reconstruct the table in Example 1 and have them note the effect on the y-value as x is increased by 1. 

Use Examples 2 and 3 to show students what happens when the constant A in the function y = A ■ b x is 
changed. 

• Compare graphs side-by-side for several values of A. 

Use Example 4 to show students what happens when the constant b in the function y = A ■ b x is changed. 

• Compare graphs side-by-side for several values of b. 
Allow students to make essential observations: 

• All exponential functions have the same basic shape: start small and grow fast. 

• At x = 0, the y-value is equal to A. Therefore, A is the y-intercept and called the initial value. Draw 
the analogy with b in the linear equation y = mx + b. 

• The larger the base, the faster the y-values will increase. 

Use Examples 6 and 7 to show students a general way to find the base b. 

Have students put their answers back in context of the problem. For example, despite the negative, the 
number x = -5 in Example 6 is useful; the interpretation is 5 years ago. 

Error Troubleshooting 

In Review Questions 1-4, have students include and negative values for x. Remind students that negative 
exponents mean division (reciprocals). 

In Review Question 6, remind students to convert the percent to a decimal. 

8.6 Exponential Decay Functions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph an exponential decay function. 

• Compare graphs of exponential decay functions. 

• Solve real-world problems involving exponential decay. 

Vocabulary 

Terms introduced in this lesson: 

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exponential decay 

asymptote 

exponentially decreasing 

half-life 

depreciation 

Teaching Strategies and Tips 

Use the introductory problem to motivate the exponential decay model. 

Suggest to students that exponential decay has the same general form as exponential growth, y = A ■ b~ x . 

• Allow students to compare and contrast them. For example, unlike exponential growth, exponential 
decay models have two forms; for instance, y = (5) and y = 5~ x . 

Students benefit from comparing graphs of exponential decay and growth functions side-by-side. Construct 
tables for each function. 



Error Troubleshooting 

Encourage students to construct tables in Review Questions 1-4. Suggest that they use zero and some 
negative numbers for x. 

In Review Question 5, remind students to convert the percent to decimal. 

8.7 Geometric Sequences and Exponential Func- 
tions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify a geometric sequence. 

• Graph a geometric sequence. 

• Solve real- world problems involving geometric sequences. 

Vocabulary 

Terms introduced in this lesson: 

geometric sequence 
common ratio 
term, n th term 
discrete, continuous 

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Teaching Strategies and Tips 

Use this lesson to show how exponential functions and geometric sequences are related. 

• In a geometric sequence, terms are found by multiplying the same constant to the previous term. 

• For exponential functions, when the input variable is increased by 1, the output variable changes by 
the value of the growth rate. 

Point out that students have two important models of growth in the real world: linear and exponential. 
Have students compare and contrast them. 

• The terms of an arithmetic sequence are said to grow "linearly." The terms of a geometric sequence 
are said to grow "exponentially." 

• The discrete model of linear growth is S n — a + nd; where a is the first term and d the common 
difference. The continuous model is y = mx + b; where b is the y-intercept and m the slope. The 
discrete model of exponential growth is S n = ar"" 1 ; where the first term is a, and r is the common 
ratio. The continuous model is y = Ab x . 

Additional Examples: 

1. Find the common ratio of each geometric sequence. 

a. 3,12,48,192,... 

b. -1,1,-1,1,-1,... 

r (\ _9 2 _2 

C. U, 4, 3, g ,. . . 

d. -10,-80,-640,-5120,... 

Tips: Have students create scatterplots of the sequences in their calculators. 

• Allow them to discover how each is related to the value of the common ratio and the sign of the first 
term. 

• Have them develop an explicit formula for each sequence. 

2. A company is offering to pay you one penny on your first day at work and each day after that your 
salary would triple. Assume you work five days a week. 

a. Fill out the tables representing your daily salary for your first two weeks at work, assuming you take 
the job. 

Monday Tuesday Wednesday Thursday Friday 

0.01 

Monday Tuesday Wednesday Thursday Friday 

b. Write an equation to model the growth of your salary. 

c. How much will you make at the end of the third week? 

3. Solve for the following. 

a. In a geometric sequence, a\ = 105 and «5 = 8505. Find r. 

b. In a geometric sequence, aj = 4096 and r = 2. Find a\. 

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Error Troubleshooting 

In Example lb and Review Questions 5 and 6, students run into difficulty because there are no consecutive 
terms that can be divided to find the common ratio. Have them use unknowns for the sequence terms 
starting with the first given term. Solve the resulting equation. See the hint at the end of Example lb. 

8.8 Problem-Solving Strategies (reprise Make a 
Table; Look for a Pattern) 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Make tables and identify patterns. 

• Solve real- world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 

compound interest 
population decrease 
intensity (loudness) of sound 

Teaching Strategies and Tips 

Remind students of the four-step problem- solving plan. 

In Example 1, students must find the compound interest formula on their own by looking for a pattern. 
Guide them through the first two years until the pattern becomes evident. 

Have students check their answers to the examples presented in this lesson by finding b, the growth or 
decay rate, instead of looking for a pattern. Have them compute (x%)A + A = ( 1 + ^) A. The method 
was presented in Examples 6 and 7 in lesson Exponential Growth Functions. 

Encourage students to construct tables and look for patterns in Review Questions 1-4. Suggest that they 
only check their work using explicit formulas. 

Error Troubleshooting 

In Review Questions 1-4, remind students to convert the percents to decimals. 



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Chapter 9 

TE Factoring Polynomials 



Overview 

In this chapter, students add, subtract, multiply, and simplify polynomials. Students learn to recognize 
special products of binomials and then move on to factoring trinomials. 

Suggested Pacing: 



Addition and Subtraction of Polynomials - 1 hr 

Multiplication of Polynomials - 1 hr 

Special Products of Polynomials - 1 hr 

Polynomial Equations in Factored Form - 1 - 2 hrs 

Factoring Quadratic Expressions - 1 - 2 hrs 

Factoring Special Products - 1 hr 

Factoring Polynomials Completely - 2 - 3 hrs 



If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 



Problem-Solving Strand for Mathematics 

In this chapter, Factoring Polynomials, our exploration is focused on the problem-solving techniques Make 
a Systematic List and Draw a Picture/Use a Model (Visualizing). 

The FlexBook repeatedly illustrates systematic lists of possible factors, particularly in lessons Factoring 
Quadratic Expressions, Factoring Special Products, and Factoring Polynomials Completely. When working 
with these lessons, point out the organizational scheme the authors have used to list all possible factors; 
this can be enlightening and give students a model from which to work. By making a systematic list, 
students begin to see patterns that can help them make better "educated guesses." An organized list will 
also help them find all possibilities. 

The use of a systematic list applies in this unit not only to the sets of possible factors and the signs of these 
factors, but also to the techniques that should be applied before deciding whether or not a polynomial is 
prime. 

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Alignment with the NCTM Process Standards 

Among the NCTM Process Standards, Communication, Connections, and Representation are most directly 
represented in this unit. By making a systematic, orderly list, students organize and consolidate their 
mathematical thinking (COM.l) and communicate their thinking coherently and clearly to peers and 
teachers (COM. 2). Through classroom discussions and sharing approaches and projects, they will analyze 
and evaluate the mathematical thinking and strategies of others (COM. 3) and will recognize and use 
connections between algebraic and geometric representations (CON.l). The tree diagrams, tables, charts, 
and matrices that students create to organize, record, and communicate mathematical ideas (R.l) are 
excellent tools for displaying data systematically. In studying the illustrations provided in the FlexBook 
and developing their own visualizations and models, students will have the opportunity to select, apply, 
and translate among mathematical representations (R.2) and to use representations to model and interpret 
physical, social, and mathematical phenomena (R.3). 

• COM.l - Organize and consolidate their mathematical thinking through communication. 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• CON.l - Recognize and use connections among mathematical ideas. 

• R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

• R.2 - Select, apply, and translate among mathematical representations to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

9.1 Addition and Subtraction of Polynomials 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write a polynomial expression in standard form. 

• Classify polynomial expression by degree. 

• Add and subtract polynomials. 

• Problem-solve using addition and subtraction of polynomials. 

Vocabulary 

Terms introduced in this lesson: 

polynomial 

term 

coefficient 

constant 

degree 

cubic term, quadratic term, linear term 

nth order term 

monomial, binomial 

standard form 

leading term 

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leading coefficient 
rearranging terms 
like terms, collecting like terms 

Teaching Strategies and Tips 

There are a large number of new terms in this lesson. Introduce new vocabulary with concrete, specific 
examples. It is also helpful to provide examples of what the new word does not mean. 

• Polynomials consist of terms with variables of nonnegative integer powers. Polynomials can have 
more than one variable. 

Examples: 

These are polynomials: 

- Vl2x 8 - x 5 + n 



x 5 + x 4 - x 3 + x 2 - 


-x+ 1 


3 

~ 7 
x 2 + y 2 




xy 




2x 2 - 4xy + 1 





These are not polynomials: 



Vx + X 


+ 2 


y 4 - 2 - x 


3.5 +x l.l 


1 
X+-- 
X 


3 


2 A " + 8 




x~ 2 + X 


- x + l 



x+ 1 



Have students explain their answers. Suggest that they use explanations such as: 
This is not a polynomial because... 

...it has a negative exponent. 

...it has a radical. 

...the power of x appears in the denominator. 

...it has a fractional exponent. 

...it has an exponential term. 

• Terms are added or subtracted "pieces" of the polynomial. 

Examples: 

The polynomial x 5 + x 4 - x 3 + x 2 - x + 1 has 6 terms. 

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The polynomial 2x 2 — 4xy + 1 has 3 terms; it is called a trinomial. 

x 2 + y 2 is a femomial because it has 2 terms. 

xy and — = are 1— term poynomials and are called monomials. 

In the polynomial -2x 5 + 7x 3 — x + 8, the 8 is a term; but neither -2, 5, 7, nor 3 are terms, —x is another 
term; but x 3 is not. 7x 3 is a term. 

• The constant term is that number appearing by itself without a variable. 

Examples: 

In the polynomial -2x 5 + 7x 3 - x + 8, the 8 is the only constant term. 

The polynomial x 2 + y 2 has no constant terms. 

• Coefficients are numbers appearing in terms in front of the variable. 

Examples: 

2x 2 - 4xy + 1. The coefficient of the first term is 2. The coefficient of the second term is -4. 

x 5 + x 4 - x 3 + x 2 — x + 1 . The coefficient of each of the terms is 1 . 

• In standard form, a polynomial is arranged in decreasing order of powers; terms with higher exponents 
appear to the left of other terms. 

Examples: 

These polynomials are in standard form: 

x 5 + x 4 - x 3 + x 2 - X + 1 
xy + x - v - 1 

These polynomials are not in standard form: 

1 - x + x 2 - x 3 + x 4 + x 5 
xy + x v - 1 

• The first term of a polynomial in standard form is called the leading term, and the coefficient of the 
leading term is called the leading coefficient. 

Examples: 

The leading term and leading coefficient of the polynomial 2x 2 - 4xy + 1 are 2x 2 and 2, respectively. 

The leading term and leading coefficient of the polynomial 9x 2 + 8x 3 + x + 1 are 8x 3 and 8, respectively. 
Remind students to write polynomials in standard form. 

• Like terms are terms with the same variable(s) to the same exponents. Like terms may have different 
coefficients. A polynomial is simplified if it has no terms that are alike. 



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Examples: 

These are like terms: 

2x 3 and -8x 3 

-xy and 17.2xy 

2x, —Ax, and V2x 

-4,tt, and V2 

These are not like terms: 

x 2 y and xy 2 

x 2 y 2 and x 2 + y 2 

The polynomial -x 3 + 3.1x 2 - 4x 2 + x - 2 is not simplified. 

• The degree of a term is the power (or the sum of powers) of the variable(s). The constant term has 
a degree of 0. The degree of a polynomial is the degree of its leading term. Encourage students to 
name polynomials by their degrees: cubic, quadratic, linear, constant. 

Examples: 

The term -8x 3 has degree 3. 

The term 7.1x 2 y 2 has degree 4. 

x 5 + x 4 - x 3 + x 2 - x + 1 is a fifth-degree polynomial. 

9x 2 + 8x 3 + x + 1 is a cubic polynomial. Remind students to write polynomials in standard form. 

Assess student vocabulary by asking them to determine all parts (terms, leading term, coefficients, leading 
coefficient, constant term) of a given polynomial and have them describe it in as many ways as they can 
(its degree, whether it is in standard form, number of variables, etc.) See Example 1-3. 

When adding or subtracting polynomials, suggest that students do so vertically. The vertical or column 
format helps students keep terms organized. 

Example: 

Subtract and simplify. 

4x 2 + 2x + 1 - (3x 2 + x - 4) 

Solution: Subtract vertically. Keep like terms aligned. 

4x 2 + 2x + 1 
- 3x 2 - x + 4 



x 2 + x + 5 



When simplifying like terms, suggest that students rearrange the terms into groups of like terms first. This 
is especially helpful in Review Questions 11, 12, and 16. See also Example 4. 

Error Troubleshooting 

General Tip: Remind students to distribute the minus sign to every term in the second polynomial when 
subtracting two polynomials. See Example 6 and Review Questions 13-16. 

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When simplifying polynomials, such as in Example 4b and Review Questions 12 and 16, remind students 
that like terms must have the same variables and exponents. 

In Example 6, remind students that to subtract A from B means B — A and not the other way around. 

Example: 

Subtract -2m 2 + 3ra 2 + Amn - 1 from -2n 2 - 7 + 2mn + 8m 2 . 

Hint: Setup the problem as —2n 2 — 7 + 2mn + 8m 2 - {—2m 2 + 3n 2 + Amn — 1). Then distribute the negative 
inside the parentheses to every term. Group like terms. 

General Tip: Some students will give the incorrect degree of a polynomial; remind students write polyno- 
mials in standard form and then look for the leading term. 

General Tip: Students can check their answers by plugging in a simple value for the variable in the original 
polynomials and simplified polynomial and check if the results have the same value. 

Example: 

Subtract -2m 2 + 3n 2 + Amn - 1 from -2n 2 - 7 + 2mn + 8m 2 . 

Solution: 

Distribute. -2n 2 - 7 + 2mn + 8m 2 - (-2m 2 + 3ra 2 + Amn - 1) 

Group like terms. -2n 2 - 7 + 2mn + 8m 2 + 2m 2 - 3n 2 - Amn + 1. 

(8m 2 + 2m 2 ) + (-2n 2 - 3« 2 ) + (2mn - Amn) + (-7 + 1) 

Answer: 10m 2 - 5n 2 — 2mn — 6 

Check. 

Let m = -1 and n = 1. 

Original: -2n 2 - 7 + 2m« + 8m 2 - (-2m 2 + 3n 2 + Amn - 1) 

- 2 • l 2 - 7 + 2 • (-1) • 1 + 8 • (-1) 2 - (-2 • (-1) 2 + 3 • l 2 + 4 • (-1) -1-1) 
-3-(-4) = l 

Simplified: 10m 2 - hn 2 - 2mn - 6 

10-(-l) 2 -5-l 2 -2-(-l)-l-6 
10 -5 + 2-6 = 1 

9.2 Multiplication of Polynomials 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Multiply a polynomial by a monomial. 

• Multiply a polynomial by a binomial. 

• Solve problems using multiplication of polynomials. 

Vocabulary 

NOT USED 

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Teaching Strategies and Tips 

Use Example 1 to introduce polynomial multiplication beginning with monomials. 

• Have students apply the product rule for exponents to each variable separately. 

• Multiply the coefficients separately. 

Use Examples 2 and 3 to show that the product of a monomial and any polynomial uses the distributive 
property. 

• Show why the distributive property works by finding the area of a rectangle in two ways. See 
illustration preceding Example 2. 

• After distributing, students use the method to multiply two monomials. 

Use Examples 4 and 5 to demonstrate the product of two binomials. There are two approaches: 

• In-line or horizontal multiplication. Use the distributive property as before. 

• Vertical multiplication. Arrange like terms in columns. Draw the analogy to the method of multi- 
plying two integers that students learned in grade school. 

Error Troubleshooting 

General Tip: Remind students to combine like terms after multiplying. Have them check their answer and 
make sure all like terms have been combined. 

9.3 Special Products of Polynomials 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find the square of a binomial. 

• Find the product of binomials using sum and difference formula. 

• Solve problems using special products of polynomials. 

Vocabulary 

Terms introduced in this lesson: 

second-degree trinomial 
square of a binomial, binomial square 
sum and difference of terms 
difference of squares 

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Teaching Strategies and Tips 

In this lesson, students learn about special products of binomials. 

• Have students learn to recognize the basic patterns. 

• In classroom examples, use colors to denote the numbers playing the role of a and b in the formulas. 

In the special formulas, point out that b is considered positive; the sign does not go with the term. 

Example: 

Square the binomial. 

(x-3) 2 

Solution: 

The minus sign tells us to use (a — b) 2 = a 2 — 2ab + b 2 . Setting a = x and b = 3 (and not -3), 

x 2 - 2(x)(3) + (3) 2 = x 2 - 6x + 9 

Error Troubleshooting 

General Tip: Students commit a very common error when they write, for example, (x + y) 2 = x 2 + y 2 ; that 
is, they distribute the exponent over addition instead of multiplication. 

• The power rule for products does not apply to sums or differences within the parentheses. In general, 
(x" +y m )P ± x np +y m P. 

• Students can learn to avoid this mistake by recalling the exponent definition. Therefore, a polynomial 
raised to an exponent means that the polynomial is multiplied by itself as many times as the exponent 
indicates. For example: 

(x + y) 2 = (x + y) (x + y) 

• See Review Questions 1-4, especially Problem 4. 

9.4 Polynomial Equations in Factored Form 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use the zero-product property. 

• Find greatest common monomial factor. 

• Solve simple polynomial equations by factoring. 

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Vocabulary 

Terms introduced in this lesson: 

factoring, factoring a polynomial 

expanded form 

factored form 

zero product property 

factoring completely 

common factor 

greatest common monomial factor 

polynomial equation 

Teaching Strategies and Tips 

Use the introduction to motivate factoring. 

• The reverse of distribution is called factoring. 

• Whereas before students were learning the direction (a + b) (x + y) => ax + bx + ay + by; they will now 
learn to "put it back together": ax + bx + ay + by => {a + b){x + y). 

• Students realize that polynomials can be expressed in expanded or factored form 

Teachers may decide to have their students pull common factors out one at a time, instead of factoring the 
GCF in one step. 

Error Troubleshooting 

In Review Questions 9 and 12-16, remind students to set the monomial factor (x,y,a, or b) equal to zero. 

• Caution students against dividing by variables. In doing so, they will lose as a solution. See also 
Example 6. 

General Tip: Check that students are using the zero-product property correctly. 
Examples: 

a. Solve for x. 

(x + 3)(x-4) = 8 

(Are students incorrectly setting each factor equal to 8?) 

b. Solve for x. 

(x + 3)(x-4)-2 = 

(Are students incorrectly setting each factor equal to 0?) 

General Tip: Remind students when factoring the GCF out of itself to leave a 1. 

For example, 6ax 2 — 9ax + 3a + 3a(2x 2 — 3x); but 6ax 2 — 9ax + 3a = 3a(2x 2 — 3x + 1). See Example 5b and 
Review Questions 3 and 15. 

General Tip: Have students check their work by expanding the factored polynomial. 

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• By checking a problem worked out as 6ax 2 -9ax+3a = 3a(2x 2 -3x), students will convince themselves 
that a 1 is missing. 

General Tip: Suggest that students look carefully over the remaining terms after having factored out the 
GCF so as to not leave any other common factors. 

9.5 Factoring Quadratic Expressions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Write quadratic equations in standard form. 

• Factor quadratic expressions for different coefficient values. 

• Factor when a = — 1. 

Vocabulary 

Terms introduced in this lesson: 

quadratic polynomial 
quadratic trinomials 

Teaching Strategies and Tips 

In this lesson, students learn to factor quadratic polynomials according to the signs of a, b, and c: 

• a = l,b > 0, c > 0. See Examples 1-4. 

Additional Examples: 
Factor. 

a. x 2 + 15x + 26. Answer: (x + 13) (x + 2) 

b. x 2 + 13x + 40. Answer: (x + 8)(x + 5) 

c. x 2 + 20x + 75. Answer: (x + 15) (x + 5) 

• a = l,b < 0,c > 0. See Examples 5 and 6. 

Additional Examples: 
Factor. 

a. x 2 - 17x + 42. Answer: (x - 14) (x - 3) 

b. x 2 -21x+90. Answer: (x-15)(x-6) 

c. x 2 - 14x + 48. Answer: (x - 6)(x - 8) 

• a = l,c < 0. See Examples 7-9. 
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Additional Examples: 
Factor. 

a. x 2 - 15x - 54. Answer: (x - 18) (x + 3) 

b. x 2 + 7x - 60. Answer: (x + 12) (x - 5) 

c. x 2 - 16x - 192. Answer: (x - 24) (x + 8) 

• a = -1. See Example 10. 

Additional Examples: 
Factor. 

a. -x 2 -4x + 60. Answer: -(x-6)(x+10) 

b. -x 2 + 14x-40. Answer: -(x-10)(x-4) 

c. -x 2 -25x-156. Answer: -(x + 12)(x + 13) 

• Allow students to infer that if c > 0(a = 1), then the factorization will be either of the form 

( + )( + ) or ( - )( - ) (same signs). If c < (a = 1), then use the form 

( - )( - ) (different signs). 

• See summary at the end of the lesson for a list of procedures and examples for each case. 

Emphasize that factoring is the reverse of multiplication. 

• Use an example such as (x + 3)(x + 7) = x 2 + lOx + 21 in which the binomials are expanded one step 
at a time to motivate factoring. 

• Demonstrate that factoring is equivalent to putting squares and rectangles back together into larger 
rectangles. 

Example: 
Multiply. 

(x + 3)(x+7). 

Solution. The diagram shows that (x + 3)(x + 7) = x 2 + 10x + 21. Observe that it also shows how to factor 

x 2 + lOx + 21. 

x+7 



+ 



area = (x+3)(x+7) 3x 21 

area = x 1 + 10x +21 

Suggest that students stop listing the possible products for c after the correct choice is evident. 

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Error Troubleshooting 

General Tip: For quadratic trinomials with a = — 1, remind students to factor -1 from every term. Remind 
students to include it in their final answer. 

9.6 Factoring Special Products 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Factor the difference of two squares. 

• Factor perfect square trinomials. 

• Solve quadratic polynomial equation by factoring. 

Vocabulary 

Terms introduced in this lesson: 

recognizing special product 
factoring perfect square trinomials 
quadratic polynomial equations 
double root 

Teaching Strategies and Tips 

Emphasize that students are reversing the special-products formulas introduced three lessons ago. 
Have students use the vocabulary: 

• a 2 — b 2 is a difference of squares. 

• (a + b)(a — b) is the product of a sum and difference. 

• a 2 + lab + b 2 and a 2 — 2ab + b 2 are perfect square trinomials. 

• (a + b) 2 and {a — b) 2 are squares of binomials. 

The key to factoring special products is recognizing the special form, but also determining what a and b 
are. 

• Recognizing perfect integer squares, for example, may be difficult to some students. Suggest that 
students break numbers down into prime factorization first. See Example 2. 

Remind students to pull out -1 and/or the GCF in a polynomial before attempting to factor it. This 
simplifies the task dramatically. 

Error Troubleshooting 

General Tip: Remind students to check their solutions by substituting each in the original equation. 
Review Question 8 is quadratic- like. Show students that x 4 = (x 2 ) 2 . 

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9.7 Factoring Polynomials Completely 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Factor out a common binomial. 

• Factor by grouping. 

• Factor a quadratic trinomial where a + 1. 

• Solve real-world problems using polynomial equation. 

Vocabulary 

Terms introduced in this lesson: 

factoring strategy 
factor completely 
recognize special products 
common monomial 
factor by grouping 

Teaching Strategies and Tips 

Have students learn the four strategies for completely factoring a polynomial provided in the introduction. 

Use Examples 3-5 to show a new factoring technique called "factoring by grouping." 

Knowing when a polynomial is completely factored may be difficult to some students. Suggest that they 
check each factor to see if it can be factored any more. For example, a polynomial such as (x 2 — 4)(x 2 + 1) 
is not completely factored. 

Factoring is easier when the leading term has a positive coefficient. Suggest that students include the 
negative as part of the GCF right in the beginning. 

Error Troubleshooting 

NONE 



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Chapter 10 

TE Quadratic Equations and 
Quadratic Functions 

Overview 

This chapter introduces students to quadratic functions and their graphs. Students solve quadratic equa- 
tions by completing the square and using the quadratic formula. They learn that the discriminant deter- 
mines the number of roots of a quadratic equation. Students identify linear, exponential, and quadratic 
equations and finally learn to choose a model. 

Suggested pacing: 

Graphs of Quadratic Functions - 1 hr 
Quadratic Equations by Graphing - 1 hr 
Quadratic Equations by Square Roots - 0.5 hrs 
Quadratic Equations by Completing the Square - 1 hr 
Quadratic Equations by the Quadratic Formula - 1 hr 
The Discriminant - 0.5 hr 

Linear, Exponential and Quadratic Models - 1 - 2 hrs 
Problem Solving Strategies: 
Choose a Function Model - 1 - 2 hrs 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

In the earlier parts of this unit, Quadratic Equations and Quadratic Functions, a great deal of emphasis is 
placed on observing patterns that emerge when various quadratics are graphed. The classic parabolic curve, 
as viewed in example one of the opening lesson, alters in its width, orientation (opening up or down), and/or 
position on the axis in subsequent examples according to discernible differences in its equation in standard 
form. And yet, very importantly, it is noted that the shape of a parabola always remains symmetric. 

From the beginning of the unit, the text urges students to look for patterns, compare alternative approaches, 
and make an informed choice between these approaches in order to solve problems efficiently. For us as 
teachers, taking time to examine connections between various methods and investigate both their visual 

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interpretations and future applications is well worth class-time discussion and focus. Initially students 
should be encouraged to compare alternative approaches to solving quadratic equations for their relative 
simplicity and efficiency. Secondly, students should be urged to make explicit connections between methods 
and the visualization each method supports. Here again we apply the adage ascribed to Polya: "Better 
to solve one problem five ways than to solve five different problems." Thirdly, students should learn that 
a method such as completing the square, though perhaps less mechanically simple than applying the 
quadratic formula, is worth learning well because it is helpful for rewriting the equations of circles, ellipses, 
and hyperbolas. 

In the formal problem-solving lesson of the chapter, Choose a Function Model, students encounter math- 
ematical modeling, a process that begins with real-world situations and arrives at quantitative solutions 
through as many refinements and adjustments as is practicable to reach acceptable results. The Review 
Questions offer yet another opportunity for students to work in small groups. Small group, collaborative 
work allows students to talk about what they do understand. If a group is confused, you might suggest 
that the students develop two or three clarifying questions that they could present to their classmates for 
discussion or clarification. 



Alignment with the NCTM Process Standards 

This chapter, with its extensive study of quadratic equations and functions, touches on each of the process 
standards of Connections, Problem Solving, and Representation. Students recognize and see connections 
among mathematical ideas (CON.l) and understand how mathematical ideas interconnect and build on 
one another to produce a coherent whole (CON. 2). This is particularly evident in the development of 
various ways to solve quadratics, finally leading to completing the square and presenting the derivation of 
the quadratic formula. Furthermore, within each lesson and especially in the Problem Solving Strategies 
lesson at the end of the chapter, real-life applications are highlighted, applying mathematics in contexts 
outside of mathematics itself (CON. 3). 

The unit is also replete with problem-solving considerations; it builds new mathematical knowledge (PS.l), 
solves problems that arise in mathematics and in other contexts (PS. 2), and applies and adapts a variety 
of appropriate strategies to solve problems (PS. 3). Especially in the lesson that focuses on mathematical 
modeling, the unit monitors and reflects on the process of mathematical problem solving (PS. 4), encour- 
aging revisions and reformulations to be certain that the real- world situation is actualized reasonably well. 
Much of the unit creates and uses representations (tables, graphs, equations) to organize, record, and com- 
municate mathematical ideas (R.l), selects, applies and translates among mathematical representations to 
solve problems (R.2), and uses representations to model and interpret physical, social, and mathematical 
phenomena (R.3). 



CON.l - Recognize and use connections among mathematical ideas. 

CON. 2 - Understand how mathematical ideas interconnect and build on one another to produce a 

coherent whole. 

CON. 3 - Recognize and apply mathematics in contexts outside of mathematics. 

PS.l - Build new mathematical knowledge through problem solving. 

PS. 2 - Solve problems that arise in mathematics and in other contexts. 

PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

PS. 4 - Monitor and reflect on the process of mathematical problem solving. 

R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

R.2 - Select, apply, and translate among mathematical representations to solve problems. 

R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 



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10.1 Graphs of Quadratic Functions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph quadratic functions. 

• Compare graphs of quadratic functions. 

• Graph quadratic functions in intercept form. 

• Analyze graphs of real- world quadratic functions. 

Vocabulary 

Terms introduced in this lesson: 

quadratic equations 

quadratic functions 

parabola 

smooth curve 

general form, standard form of a quadratic function 

coefficients 

orientation 

dilation 

vertical, horizontal shift 

x-intercepts of a parabola 

vertex of a parabola 

symmetric, line of symmetry 

quadratic expression 

intercept form 

Teaching Strategies and Tips 

Point out that students frequently come into contact with parabolas. 

• The path of any projectile is part of a parabola. 

• Cross-sections of satellite dishes and flashlight mirrors are parabolas. 

• See introduction for more examples. 

Draw several parabolas and lead students in discovering the essential features: 

• Symmetry 

• A maximum of two x-intercepts 

• Vertices 

• None of these features are shared by the other graphs students have been studying 

Use a table of values to show why the parabola has a U-shape. 

• Allow students to observe the effect of squaring on negative numbers. 
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• Use a basic quadratic function such as y = x 2 . 
When graphing quadratic functions, suggest that: 

• Sketches do not need to be perfect, but curves should be U-shaped rather than V-shaped. In general, 
join points with a smooth curve rather than with segments. 

• Plot more points until the familiar curve is in view. More points are necessary to draw a parabola 
accurately than to graph lines, since a parabola is curved. See Example lc. 

• Knowing the vertex cuts down on the number of points to be plotted, because that is where the 
parabola opens. Advise students to choose at least one point on either side of the vertex. 

• Not all points in a table must be plotted. This is especially true for points with large y-values which 
can make a y-axis too big. See Examples la and lc. 

Use Example 2 to introduce intercept form. 

Additional Examples: 

Find the x- intercepts and vertices of the following quadratic functions. 

a. y = x 2 + 3x - 10 

Solution: Write the quadratic function in intercept form by factoring the right side. 

y = x 2 + 3x - 10 = (x + 5)(x - 2) 

Set y = 0. So, the x-intercepts are (-5,0) and (2,0). The x-value of the vertex is halfway between the 
two x-intercepts: 

=»±*— 16 

2 

To the y-value, plug in the x-value into the original equation: 

y = (-1.5) 2 + 3(-1.5) - 10 = -12.25 

The vertex is (-1.5,-12.25). 

b.y = -2x 2 + 56x + 120 

Answer: The x-intercepts are (-2,0) and (30,0) and vertex is (14,512). 

c. y = -x 2 + 15x - 36 

Answer: The x-intercepts are (12,0) and (30,0) and vertex is (7.5,20.25). 

Point out that the vertex lies on the line of symmetry; therefore, x-intercepts are reflections of each other 
and the vertex is halfway between them. 

By analyzing the effects of the coefficients a and c in v = ax 2 + c, teachers can introduce the basic 
transformations: shift, stretch, and flip. Ask: 

• What makes the graph of the parabola open up? Down? 

• What makes a parabola wider? Narrower? 

• How can the vertex of one parabola be placed higher than another? 

Use graphing calculators to: 

• Find intercepts, vertices, and points of intersection to any degree of precision. 

• Compare several quadratic functions simultaneously. 

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Error Troubleshooting 

NONE 

10.2 Quadratic Equations by Graphing 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify the number of solutions of quadratic equations. 

• Solve quadratic equations by graphing. 

• Find or approximate zeros of quadratic functions. 

• Analyze quadratic functions using a graphing calculator. 

• Solve real-world problems by graphing quadratic functions. 

Vocabulary 

Terms introduced in this lesson: 

roots 

zeros 

distinct solutions 

double root 

no real solutions 

Teaching Strategies and Tips 

Emphasize the connection between algebra and geometry: 

• Finding the roots of a quadratic function (algebra) is equivalent to finding the x-intercepts of a 
parabola (geometry). 

• Have students use correct vocabulary: equations have roots or zeros; graphs have x-intercepts. 

Use Example 4 to explore the graph of an equation using a graphing calculator. 

• In graph mode, use the cursor to scroll over the x-intercepts or vertex to find an approximate value 
of each. Approximations can be improved by zooming in. 

• In graph mode, from the CALC menu, use ZERO and MAXIMUM (MINIMUM) to find an x-intercept 
and vertex, respectively. 

• Use the built-in table to find an x-intercept (look for the row with y = 0) or a vertex (scroll through 
the table values until the y values change from increasing to decreasing; or vice- versa). By changing 
the table's step size, the approximation can be made better. 

• Emphasize approximating roots of an equation by reading a graph is an essential skill; real-world 
equations rarely factor over the integers. 

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In Example 5, encourage students to explore the equation on a graphing calculator using the WINDOW 
menu. By changing parameters XMIN, XMAX, YMIN, and YMAX, students learn to find an appropriate 
display for any graph. 

Have students interpret their answers. In Example 5, the two roots indicate the two times when the arrow 
is on the ground. 

General Tip: In Review Questions 1-12, have students check their answers with a graphing calculator for 
added practice. 

Error Troubleshooting 

In Review Question 6, remind those students inputting the equation into a calculator to use proper syntax. 
The following are equivalent: 

. yi = (l/2)x A 2-2x + 3 
. yx = x A 2/2-2x + 3 
. yi = (x A 2)/2-2x + 3 

10.3 Quadratic Equations by Square Roots 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve quadratic equations involving perfect squares. 

• Approximate solutions of quadratic equations. 

• Solve real-world problems using quadratic functions and square roots. 

Vocabulary 

Terms introduced in this lesson: 

quadratic equations involving perfect squares 
positive, negative square root 
approximate solutions 
free fall 

Teaching Strategies and Tips 

Remind students that: 



• Vx 2 = |x| and is called the principal square root. 

• |x| = k implies x = +k. 

• Conclude that x 2 = c has two solutions when c > 0. See Examples 1-3. 

• Point out that square roots of negative numbers are not real numbers. Therefore, x 2 = c has no 
solutions when c < 0. 

Specify in advance the number of decimal places required of students in Examples 6 and 7 and Review 
Questions 18-22. 

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Error Troubleshooting 

In Example 2, show students that J-§- = 3 and J-^ = 5. More square-roots of fractions can be found in 
Review Questions 4-6 and 8. 

General Tip: Remind students after square-rooting both sides of an equation to include the ±. 

10.4 Quadratic Equations by Completing the Square 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Complete the square of a quadratic expression. 

• Solve quadratic equations by completing the square. 

• Solve quadratic equations in standard form. 

• Graph quadratic equations in vertex form. 

• Solve real-world problems using functions by completing the square. 

Vocabulary 

Terms introduced in this lesson: 

completing the square 

perfect square trinomial 
quadratic equations in standard form 
quadratic equations in vertex form 
parabola turns up, turns down 

Teaching Strategies and Tips 

Use Example 1 to introduce completing the square. 

• Remind students of binomial expansions to help them understand how a constant term turns the 
expression into a perfect square trinomial: 

(x±a) = x ± 2ax + a 

• Point out that the leading coefficient is 1. 

• In Example 3, a £ 1. Students learn to factor a from the whole expression before completing the 
square, whether or not the terms are multiples of a. Complete the square of the resulting expression 
in parentheses. 

Give completing the square geometrical meaning. 

• Use squares and rectangles for each term of the expression. 

• See paragraph preceding Example 4. 

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• Have students make up a quadratic expression and ask them to complete the square in the geometrical 
interpretation as an assignment. 

There are several reasons to have students learn completing the square: 

• It is used to derive the quadratic formula. 

• Quadratic equations can be rewritten in vertex form. 

• Equations of circles can be rewritten in graphing form. 

• Necessary in calculus. 

Emphasize that completing the square finds roots 

• Regardless of whether the roots are integers, rational or irrational numbers. 

• Without having to list all the cases, unlike factoring. 

Example: 

Solve the following quadratic equation: 

2x 2 + x - 3 = 

Solution: Although 2x 2 + x — 3 = (2x + 3)(x — 1), and therefore x = — f and x = 1; we show that completing 
the square results in the same solutions. 

Rewrite: 

2x 2 + x = 3 

Divide by the leading coefficient; this results in an equation with a = 1. 

(i: 

Add the constant to both sides of the equation: 

' Hi) 

Factor the perfect square trinomial and simplify. 

K)' 



■ v2 + , i)H 



-MIMi) ' " 



1\ 2 _ 3 1 
" 2 + 16 

1\ 2 24 1 



hi) =- + 



16 16 
1\ 2 25 

" ! -' -is 



K) 



Take the square root of both sides: 



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/ 1\ 2 /25 

1 5 

x + - = +- 
4 4 

1 5 
-1±5 

-1 + 5 



4 

-1-5 
x - 



1 



4 2 

Answer: x = -| and x = 1 as expected. 

Use Examples 7-9 to show how completing the square helps in graphing quadratic functions. 

Error Troubleshooting 

Dividing by the leading coefficient in Review Questions 7, 8, and 14 results in a fractional b coefficient. 
Remind students that dividing b by 2 is equivalent to multiplying b by 1/2. 

Additional Example: 

Solve the quadratic equation by completing the square: 

3x 2 + 5x - 3 = 

Hint: Rewrite the equation: 

3x + 5x = 3. 



And divide by the leading coefficient: 



2 5 
X + -x = 1 



To find the constant that must be added to both sides of the equation, multiply 5/3 by 1/2 and then 
square: 

'5 1\ 2 /5^ 2 



(■ -0 



General Tip: Remind students to rewrite the vertex form of a quadratic equation with minus signs to find 
h and k correctly. See Examples 7-9. 

Additional Example: 

Find the vertex of the parabola with equation: 

y + 3 = -2(x-5) 2 
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Solution: Rewrite the equation with minus signs: 

y-(-3) = -2(x-5) 2 

The vertex is at (h,k), where h = -3 and k = 5. 

General Tip: Remind students that the leading coefficient must be 1 before completing the square. They 
will need to divide or factor to accomplish this. 

General Tip: Suggest that students rewrite equations in standard form before completing the square. 

In Examples 10 and 11 and Review Questions 25 and 26, have students round in the last step. 

10.5 Quadratic Equations by the Quadratic For- 
mula 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve quadratic equations using the quadratic formula. 

• Identify and choose methods for solving quadratic equations. 

• Solve real-world problems using functions by completing the square. 

Vocabulary 

Terms introduced in this lesson: 

quadratic formula 

roots, solutions to a quadratic equation 

Teaching Strategies and Tips 

Emphasize that the quadratic formula comes from completing the square of a general quadratic equation. 

• Point out that half the coefficient of x squared can be found by multiplying by 1/2: 

b l\ 2 _/_^ x2 
a 2/ \2a 

• Remind students to find common denominators before simplifying the right side of the equation. 

Have students rewrite each quadratic equation in standard form first. See Example 3. 

Use Example 4 to show students how one goes about choosing a solving method. 

Teachers are encouraged to use quadratic equations with non-integer coefficients — decimals and frac- 
tions — as additional practice problems. 

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Error Troubleshooting 

General Tip: Some students may use the values of a, b, and c based upon the order in which the terms 
were given. Students will assume that a is the first coefficient, b the second, and c the last. 

• Remind students rewrite each quadratic equation in standard form first. See Example 3. 
General Tip: Have students simplify one step at a time when using the quadratic formula: 

• Substitute. 

• Simplify under the radical. 

• Determine the square root. 

• Simplify the numerator. 

• Simplify the denominator. 

• Divide. 

General Tip: Some common mistakes associated with the quadratic formula are: 

• Not using the minus sign that goes with a coefficient. 
Example: 3x 2 + 7x - 4 = 0. Use a = 3,b = 7, and c = -4.(c ± 4) 

• Losing the minus sign on —b. 

Example: 3x 2 — 7x + 4 = 0. Use a = 3,b = -7, and c = 4. 

x= - ( - 7)± ^ 3 7 ; 2 - 4(3)(4 i and notx= - 7± V ( -g ) - 4(3 M 

• Canceling a factor of the denominator with — b only. Remind students that in order to cancel a factor 
from the denominator, it must be a common factor in every term in the numerator. Unless that 
factor comes out of the radical, then the factor in the denominator cannot be canceled. 

Example: For a = 3, b = 6, and c = -1, x = igi^gMEg = z^JM ^ x= d^M. 

• Rounding at steps other than the last step. 

10.6 The Discriminant 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find the discriminant of a quadratic equation. 

• Interpret the discriminant of a quadratic equation. 

• Solve real-world problems using quadratic functions and interpreting the discriminant. 

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Vocabulary 

Terms introduced in this lesson: 

discriminant 
double root 
nature of solutions 
distinct real solutions 
non-real solutions 
rational number solutions 
irrational number solutions 

Teaching Strategies and Tips 

In this lesson, students learn that each quadratic equation is associated with a number whose sign tells 
how many solutions there are to that equation. 

• Emphasize that the equation does not have to be solved. 

Point out that if the discriminant is not a perfect square, then the solutions are irrational numbers. See 
Example 3. 

Use Examples 5-7 to illustrate how useful interpreting the discriminant in context of the problem can be. 

Error Troubleshooting 

General Tip: Remind students to use care when substituting values into the discriminant. 

10.7 Linear, Exponential and Quadratic Models 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Identify functions using differences and ratios. 

• Write equations for functions. 

• Perform exponential and quadratic regressions with a graphing calculator. 

• Solve real-world problems by comparing function model. 

Vocabulary 

Terms introduced in this lesson: 

ratio of the differences 
constant ratio of differences 
exponential, quadratic regression 

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Teaching Strategies and Tips 

Emphasize that students should always begin a regression by drawing the scatter plot of the data. This 
allows them to determine which function of best fit is most appropriate, if any. 

Point out: 

• A regression is a curve that comes as close as possible to all the data points without being another 
type of function. 

• Regressions do not necessarily pass through data points. Therefore, y-values will be different from 
those in the data set. 

Remind students to check in general the differences in x— values that they are constant. 

Error Troubleshooting 

In Review Questions 4-6 and 12, remind students to calculate the differences of the differences to see if a 
quadratic trend exists. Students often calculate only the first set of differences. 

10.8 Problem Solving Strategies: Choose a Func- 
tion Model 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Read and understand given problem situations. 

• Develop and use the strategy: Choose a Function Model. 

• Develop and use the strategy: Make a Model. 

• Plan and compare alternative approaches to solving problems. 

• Solve real-world problems using selected strategies as part of a plan. 

Vocabulary 

Terms introduced in this lesson: 
logistic regression 

Teaching Strategies and Tips 

The lesson presents several mathematical models arising from real-world applications in the examples. 

Encourage students to read and understand the modeling process presented as a flow chart in the intro- 
duction. 

In each of the examples, present and compare several regressions side-by-side. Allow students to see how 
accurate each is. 

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Error Troubleshooting 

General Tip: Students rush to calculate the regression not stopping to consider its appropriateness. A 
scatter plot should always be consulted first. 

General Tip: Have students check the data points they have inputed into a calculator twice. 

General Tip: On the TI graphing calculators, students should be using LinReg{ax + b) or LinReg{a + bx)to 
perform linear regressions and not LnReg. 



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Chapter 11 

TE Algebra and Geometry 
Connections; Working with Data 

Overview 

In this chapter, students are introduced to the square root function and learn how shifts, flips, and stretches 
change the shape of the graph. Students use the properties of radicals to solve radical equations and 
problems involving the Pythagorean theorem, distance and midpoint formulas. The chapter ends with an 
introduction to descriptive statistics and graphical displays. 

Suggested pacing: 

Graphs of Square Root Functions - 1 - 2 hrs 

Radical Expressions -2-4 hrs 

Radical Equations - 1 hr 

The Pythagorean Theorem and Its Converse - 1 hr 

Distance and Midpoint Formulas - 1 hr 

Measures of Central Tendency and Dispersion - 1 hr 

Stem-and-Leaf Plots and Histograms - 1 hr 

Box-and- Whisker Plots - 1 hr 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

Along with the four-step problem-solving plan presented in the text, students are prompted to reflect back 
on strategies, evaluate their effectiveness and analyze their usefulness for future problems. This chapter, 
Algebra and Geometry Connections; Working with Data, continues to promote such an analysis. 

Recognizing patterns, drawing connections, and considering extraneous solutions are all featured. When 
dealing with extraneous solutions, point out to students where in the procedures the potential for a false 
solution has been introduced. This awareness can prepare them for more sophisticated mathematics ahead. 

Encourage students to think about everyday applications of the Pythagorean Theorem (students might 
interview carpenters, plumbers, architects, and/or structural engineers, for example) and emphasize that 

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the distance and midpoint formulas are extensions of the basic Pythagorean theorem and "taking an 
average" respectively. Similarly, with Stem-and-Leaf Plots, Histograms, and Box-and- Whisker Plots, be 
sure students understand the focus and strengths of each style of graph. The point with each of these tools 
is to represent and/or be able to interpret data in ways that make sense and communicate clearly. 

Alignment with the NCTM Process Standards 

This chapter, focused on algebra and geometry connections and working with data, aligns with many of 
the NCTM process standards. Throughout students are asked to recognize and use connections among 
mathematical ideas (CON.l), to appreciate how mathematical ideas interconnect and build on one another 
to produce a coherent whole (CON. 2) and to recognize and apply mathematics in contexts outside of 
mathematics (CON. 3). When dealing with radical equations and extraneous solutions, students apply and 
adapt a variety of appropriate strategies to solve problems (PS. 3) and monitor and reflect on the process of 
mathematical problem solving (PS. 4). In working with the Pythagorean theorem and its converse, students 
use representations to model and interpret physical and mathematical phenomena (R.3). In applying the 
distance and midpoint formulas students select, apply, and translate among mathematical representations 
to solve problems (R.2). 

In the lessons which produce and analyze data communication tools, students create and use representation 
to organize, record, and communicate mathematical ideas (R.l); communicate their mathematical thinking 
coherently and clearly to peers, teachers, and others (COM. 2); analyze and evaluate the mathematical 
thinking and strategies of others (COM. 3); and use the language of mathematics to express mathematical 
ideas precisely (COM. 4). 

• COM. 2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and 
others. 

• COM. 3 - Analyze and evaluate the mathematical thinking and strategies of others. 

• COM. 4 - Use the language of mathematics to express mathematical ideas precisely. 

• CON.l - Recognize and use connections among mathematical ideas. 

• CON. 2 - Understand how mathematical ideas interconnect and build on one another to produce a 
coherent whole. 

• CON. 3 - Recognize and apply mathematics in contexts outside of mathematics. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• PS. 4 - Monitor and reflect on the process of mathematical problem solving. 

• R.l - Create and use representations to organize, record and communicate mathematical ideas. 

• R.2 - Select, apply, and translate among mathematical representations to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

11.1 Graphs of Square Root Functions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Graph and compare square root functions. 

• Shift graphs of square root functions. 

• Graph square root functions using a graphing calculator. 

• Solve real-world problems using square root functions. 

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Vocabulary 

Terms introduced in this lesson: 

increases, decreases 

flip 

shift 

stretch 

transform 

Teaching Strategies and Tips 

Use Example 1 to introduce the basic shape of the square root function. 

Use the tables in the examples to show students the square root function's behavior numerically: 

• Why it is undefined on some intervals. 

• That it is everywhere increasing. 

• That it rises relatively slowly. 

• That the square root of a fraction is greater than the fraction. Numbers in the interval (0,1) are 
smaller than their square roots; x < V* f° r x i n t ne interval (0, 1) and x > yfx for x in the interval 
(l,oo). 

Use the graphs in the examples to make the following observations: 

• The square root graph is half a parabola lying sideways. 

Emphasize finding the domain of the square root function before making a table. 

• When the expression under the square root is negative, table values will be undefined; and the graph 
corresponding to the interval will be empty. 

Use Examples 2-10 to motivate transformations: 

• Shifts, stretches, and flips allow graphing without constructing a table of values. 

• Teachers are encouraged to use several examples to illustrate the effect of each constant on the graph. 

Additional Examples: 

Graph the following functions using transformations of the basic graph y = yfx. 

a. y = - ypc 

Hint: Flip about the x-axis. 

b. y = V~* 

Hint: Flip about the y-axis. 

Remark: Student often claim that the whole function is undefined because of the negative under the radical. 
Point out that the domain "flips" to negative numbers. 

c. y = 2 Vx 

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Hint: Stretch in the vertical direction by a factor of 2: y-values are multiplied by 2. 

d. y= y[2x 

Hint: Point out that y = V2x = V2 yfx . 

e. y = ypc + 2 

Hint: Shift the graph up by 2: y-values are increased by 2. 

f. y= V^T2 

Hint: Shift the graph left by 2: x— values are decreased by 2. 

g- y = - V* + i 

Hint: Point out that the transformations are in the same direction, y-values are reflected across the x-axis 
(parallel to the y-axis) and then shifted vertically (parallel to the y-axis), in that order. Therefore, the 
correct sequence is to flip and then shift. 

h. y = yj-x + 1 

Hint: Have students consider what happens to an input x and do the transformations in the opposite order. 
Therefore, shift left, then reflect across the y-axis. 

Additional Example: 

Graph the following function using shifts, flips, and stretches. 

a. y = 4 + 2V2-x 

Solution: View y = 4 + 2 V2 - x as a combination of transformations of the basic square root graph y = -\fx. 

Start with the simpler equation: y = V2 - x. If we follow an input x, then it first gets multiplied by -1 
and then increased by 2. How do the graphs of y = y[x and y = V2 - x compare? y = V2 - x is a shift of 
y = \[x two units LEFT and then flipped across the y-axis. Note that the transformations happen in the 
opposite order in which an input x gets operated on. 

To graph y = 2 V2 - x we multiply the y-values of y = V2 - x by 2 to obtain a vertically stretched curve. 
Finally, to obtain the graph of y = 4 + 2 V2 - x, shift the graph of y = 2 V2 - x four units vertically. 

Encourage students to keep a list of functions they have studied so far. Include a few examples of each 
and their graphs. For example: 

• Linear: f{x) = mx + b 

Examples: f(x) = x, f(x) = -x, f(x) — x+1, f(x) = 2 

• Exponential: f(x) = a ■ b x 
Examples: f{x) = 2\ /(*) = 2~ x , f{x) = -2 X 

• Quadratic: f(x) = ax 2 + bx + c 
Examples: f(x) = x 2 , f(x) = -x 2 , f(x) = x 2 + 1 

• Square root: f(x) = a ^Ibx + c + d 

Examples: f(x) = ^pc, f(x) = y/x + 1, f(x) = ^pc+ 1, f(x) = — V* 

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Error Troubleshooting 

General Tip: Students may not recognize y = V~* as a valid function at first, stating that the square root 
of a negative is undefined. Explain that the function's domain is defined. 

General Tip: Have students find the domain of the function they are graphing on a calculator first. This 
will help find an appropriate window for the graph. 

In Example 12 and Review Questions 14-18, state beforehand the number of decimal places required of 
students when rounding. 

11.2 Radical Expressions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use the product and quotient properties of radicals. 

• Rationalize the denominator. 

• Add and subtract radical expressions. 

• Multiply radical expressions. 

• Solve real-world problems using square root functions. 

Vocabulary 

Terms introduced in this lesson: 

radical sign 
even roots, odd roots 
simplest radical form 
rationalizing the denominator 

Teaching Strategies and Tips 

In this lesson, students learn that: 

• Radicals reverse the operation of exponentiation. 

• The index determines the kind of root. 

• The square root is the only index which is not explicitly written but understood. 

Use Example 1 to point out that even and odd indices handle negatives differently. 
Additional Examples: 

a. V-64 is not a real number. 

b. V-64 is defined and evaluates as -4. 

Use Example 2 to motivate rational exponents. 

• One way to justify that i[a = a 1 '" is to use remind students of the power rule: \x m \ = x mn . 
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• Let m = g and n = 2: 1x2 J = x 1 — x. 

• Therefore, x 1 ' 2 is a number that when squared equals x. Therefore, x 1 ' 2 = -\fx. 

• A similar argument holds in general for any index: ifa = a 1 '". 

i 

• Using the power rule again: a m ' n = a m '» = la'") = Va"*. 
. Therefore, 1/a" = a n l n . 

Have students state the radical properties in words. This can help students learn the rules: 

• The product rule for radicals: The square root of the product is the product of the square roots. 

• The quotient rule for radicals: The square root of the quotient is the quotient of the square roots. 

Rationalizing the denominator: 

• Remind students to multiply the numerator and denominator by the radical expression. "What you 
do to the top you do to the bottom." 

• Point out that rationalizing the denominator is essentially multiplying by 1; therefore, the value of 
the original rational expression does not change. 

• Have students seek a radical expression that when multiplied with the denominator results in a 
perfect power. 

• In the case when the denominator contains two terms, one being a radical, a good choice for the 
rationalization is an expression whose product is a difference of squares. 

• See Review Questions 26-33. 

Encourage students to leave their answers in radical form unless otherwise specified. 

• Radical form is an exact answer. 

• If a decimal is needed, the final radical can be rounded. 

Have students simplify all radicals to simplest form to ensure that all possible like terms in the expression 
are combined. 

For students having a difficult time adding and simplifying radical expressions, draw the analogy with 
combining like terms in variable expressions. For example, the expressions -2x + 7x and -2 yb + 7 V5 are 
essentially the same. 

Teachers are encouraged to be specific about when a radical is in simplified form: 

• No fractions occur in the radicand. 

Example: The expression x 2 y.J| can be simplified to ^-. 

• No radicals are present in the denominator of a fraction. 

Example: The expression -j= can be simplified to — or x~2 . Decide whether to include negative exponents 
in simplified form. 

• The index of a radical and the exponents on any expressions in the radicand do not have common 
factors. 

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Example: The expression Vjc 2 can be simplified to ^fx. 

• The exponents on any expressions in the radicand are less than the index. 
Example: The expression V* 9 can be simplified to x \x^. 

• The resulting expression has as few radicals as possible. 
Example: The expression y5 + V20 can be simplified to 3 V5. 

Error Troubleshooting 

In Review Questions 9-16 have students look for the highest possible perfect squares, cubes, fourth powers, 
etc. as indicated by the index of the radical. Suggest that they use factor trees as guides. 

In Review Questions 14-16, have students treat the constants and variables separately. 

General Tip: Remind students that when adding and subtracting radical expressions to combine only like 
radical terms (the same expression under the radical sign). This is analogous to combining like terms in 
variable expressions. 

In Review Question 25, remind students to multiply the numbers outside the radical sign and the numbers 
inside the radical sign separately. Use the rule: a \b ■ c yd = ac ybd. 

11.3 Radical Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Solve a radical equation. 

• Solve radical equations with radicals on both sides. 

• Identify extraneous solutions. 

• Solve real-world problems using square root functions. 

Vocabulary 

Terms introduced in this lesson: 

radical equation 
radical expression 
extraneous solutions 

Teaching Strategies and Tips 

Up to this point, students have been solving linear and quadratic equations. In this lesson, they now look 
at solving radical equations. 

The following steps are used to solve radical equations: 
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• Isolate a radical. 

• Square both sides of the equation (or use another appropriate power). 

• Solve the new polynomial equation now free of radicals. 

• Check answers in the original equation. 

Use Example 2 to show that radical equations containing radicals of any index - not just square roots - 
can be solved. 

• The steps are identical except for a change in the power that each side of the equation is raised to. 
Use Example 4 to show that radical equations containing more than one radical expression can be solved. 

• Isolate the most complicated radical expression and raise the equation to the appropriate power. 

• Repeat the process until all radical signs are eliminated. In Example 4 and Review Questions 11-16, 
students must square both sides, twice. 

Some radical equations can be made easier by reducing all terms by a common factor. 

• This should occur at the beginning of the problem or after the step when both sides have been raised 
to a power. 

Example: 

Find the real solutions of: 



4 V2x + 1 = 12-8V* 



Hint: Divide by 4 first. 



V2jc + 1 = 3 - 2 V3 

After squaring both sides and simplifying, the equation is: 

12V^ = 2x + 8 

Divide again, this time by 2 before squaring both sides. 

Point out that an equation such as Vx + 5 = -3 can be readily answered as not having any real solutions. 

Error Troubleshooting 

In Review Questions 13-16, remind students to isolate a radical first. By not isolating, some radical will 
always remain in the equation and can even make the equation more complicated. 

General Tip: Remind students to raise each side of the equation to the appropriate power, rather than 
term by term. Encourage students to use parentheses for each side. 

General Tip: Have students always check their answers for extraneous solutions. 

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11.4 The Pythagorean Theorem and Its Converse 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Use the Pythagorean theorem. 

• Use the converse of the Pythagorean theorem. 

• Solve real-world problems using the Pythagorean theorem and its converse. 

Vocabulary 

Terms introduced in this lesson: 

Pythagorean theorem 

hypotenuse 

legs 

converse 

Teaching Strategies and Tips 

Students learn that the Pythagorean theorem relates the lengths of the sides of a right triangle to each 
other. 

Students also learn that the Pythagorean theorem has a converse. 

• It can be used to verify that a triangle is a right triangle. 

• If it can be shown that the three sides of a triangle make the equation a 2 + b 2 = c 2 true, then the 
triangle is a right triangle. 

Remind students of the following few prerequisites from geometry: 

• A right triangle is one that contains a 90 degree angle. 

• The side of the triangle opposite the 90 degree angle is called the hypotenuse. 

• The sides of the triangle adjacent to the 90 degree angle are called the legs. 

• The longest side of a right triangle is the hypotenuse. 

Encourage students to say the Pythagorean theorem in words. 

Point out that knowing the value of two variables in the Pythagorean theorem is sufficient for determining 
the third. See Example 4. 

Suggest that students discard negative solutions obtained from radical equations in this lesson since the 
lengths of sides of triangles are nonnegative. See Example 5. 

Specify how many decimal places are required of students when rounding. 
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Error Troubleshooting 

General Tip: Remind students to set the value of c as the length of the hypotenuse; the values for a and 
b can be switched. 

In Review Question 18, assume that the hypotenuse is also the diameter of the circle. 

11.5 Distance and Midpoint Formulas 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Find the distance between two points in the coordinate plane. 

• Find the missing coordinate of a point given the distance from another known point. 

• Find the midpoint of a line segment. 

• Solve real-world problems using distance and midpoint formulas. 

Vocabulary 

Terms introduced in this lesson: 

distance 

equidistant 

midpoint 

Teaching Strategies and Tips 

Use Examples 1 and 2 to show how the Pythagorean Theorem is used to derive the distance formula. 

• Teachers are encouraged to use a picture in the derivation. 

Use Examples 3-5 and Review Questions 7, 8, and 15-20 as thinking problems. 

• Draw pictures to help. 

• Contrast these problems with the mechanical exercises of Example 2 and Review Questions 1-6. 

Point out that because of the squares in the distance formula, the order in which the x-values (and the 
order of the y-values) are plugged in does not matter. 

Point out that the midpoint of a segment is found by taking the average values of the x— and y-values of 
the endpoints. 

In Example 9, suggest that students express their answers in radical form. 

Error Troubleshooting 

General Tip: For points with negative coordinates, remind students about the minus sign in the distance 
formula. 

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Example: 

Find the distance between the two points. 

(-2,5) and (3,-8). 

Hint: Plug the values of the two points into the distance formula; notice that parentheses were used around 

-8. 

rf => /(-2-3)2 + (5-(-8))2 

11.6 Measures of Central Tendency and Disper- 
sion 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Compare measures of central tendency. 

• Measure the dispersion of a collection of data. 

• Calculate and interpret measures of central tendency and dispersion for real-world situations. 

Vocabulary 

Terms introduced in this lesson: 

measure of central tendency 

average 

data set 

mean 

median 

mode 

sum of values, number of values 

ordered list 

outlier 

dispersion 

variance 

standard deviation 

Teaching Strategies and Tips 

Students learn that "average" has a general meaning: 

• Averages describe the general characteristics of a group. 

• Mean, median, and mode are examples of averages. 

Use Examples 1 and 2 to introduce the mean. 
Additional Examples: 

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Find the mean of the given data. 

a. The students in Mr. Peterson's math class took the AP Statistics exam. Their math scores are: 

3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3 

Answer: 3.18 

b. The weights, in ounces, of several cookies taken from the same package are: 

0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91 

Answer: 0.92 

c. The precipitation, in mm, for the city of Townville in the month of October is: 

142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 

Answer: 38.37 

Use Examples 3 and 4 to introduce the median. 

Additional Examples: 

Find the median of the given data. 

a. The students in Mr. Peterson's math class took the AP Statistics exam. Their math scores are: 

3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3 

Answer: 3 

b. The weights, in ounces, of several cookies taken from the same package are: 

0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91 

Answer: 0.92 

c. The precipitation, in mm, for the city of Townville in the month of October is: 

142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 

Answer: 12.185 

Use Example 5 to introduce the mode. 

Additional Examples: 

Find the mode of the given data. 

a. The students in Mr. Peterson's math class took the AP Statistics exam. Their math scores are: 

3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3 

Answer: 3 and 4. 

b. The weights, in ounces, of several cookies taken from the same package are: 

0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91 

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Answer: None. 

c. The precipitation, in mm, for the city of Townville in the month of October is: 

142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 

Answer: 0.00 

Use Examples 6-8 to introduce measures of dispersion. 

Additional Examples: 

Find the range, variance, and standard deviation of the given data. 

a. The students in Mr. Peterson's math class took the AP Statistics exam. Their math scores are: 

3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3 

Answer: 

range = 4 
s 2 = 1.109 
s = 1.053 

b. The weights, in ounces, of several cookies taken from the same package are: 

0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91 

Answer: 

range = 0.12 
s 2 = 0.0018 
s = 0.0430 

c. The precipitation, in mm, for the city of Townville in the month of October is: 

142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 
Answer: 

range = 142.2 
s 2 = 2601 
s = 51 

In Example 7, students learn to separate the components of the standard deviation formula into manageable 
pieces. 

• Suggest that students reconstruct such a multi-column table for Review Question 2. 

• Suggest that students use a calculator to only to check their calculations. 

Encourage students to provide the kind of explanations shown in Example 9 for Review Questions 4 and 
5. Ask: 

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• What does a larger mean imply? 

• What does a larger standard deviation imply? 

Emphasize that the appropriateness of an average depends on the shape of the distribution of the data. 

• Use the mean for tightly clustered data sets. 

• Use the median for data sets with a lot of spread. 

• Use the mode when there is no apparent pattern in the distribution. 

Error Troubleshooting 

General Tip: Remind students to order the data set when computing the median and the range. 
General Tip: Suggest that students transfer data to calculator carefully. 

• Have students put a line through the data points as they get transferred. 

• Have students count the number of data points in both lists to prevent omissions. 

11.7 Stem-and-Leaf Plots and Histograms 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Make and interpret stem- and- leaf plots. 

• Make and interpret histograms. 

• Make histograms using a graphing calculator. 

Vocabulary 

Terms introduced in this lesson: 

visual representation of data 
stem-and-leaf plot 

stem, leaf 

ordered stem-and-lead plot 

frequency 

histogram 

bins 

continuous data 

round 

truncate 

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Teaching Strategies and Tips 



In the previous lesson, students learned that the appropriateness of an average depends on the shape of 
the distribution of the data. 

• Plotting data is essential and should come naturally as a first step in data analysis. 

• In this lesson, students learn to group and visualize data using stem-and-leaf plots and histograms. 
See Examples 1 and 2, respectively. 

Emphasize the similarity between histograms and stem-and-leaf plots - a stem-and- leaf plot resembles a 
histogram on its side. 

Teachers are encouraged to present several examples of stem-and-leaf plots consisting of different kinds of 
data sets, such as: clustered data that comes as three-digit numbers, data spread out coming in three- 
digits, data consisting of numbers with 1 or two decimal places, and data sets with decimals between 
and 1. 

Additional Examples: 

Arrange the data into a stem-and-leaf plot. 

a. The students in Mr. Peterson's math class took the AP Statistics exam. Their math scores are: 

3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3 

b. The weights, in ounces, of several cookies taken from the same package are: 

0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91 

c. The precipitation, in mm, for the city of Townville in the month of October is: 

142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 

Teachers are encouraged to show side-by-side histograms for the same data set as patterns can be more or 
less apparent with different number of bins. 

• By increasing or decreasing bin-widths, students see how the shape of the distribution changes for 
better or worse. 

• Some bin-widths can be random-looking or even misleading. 

Error Troubleshooting 

Remind students in Review Questions 1 and 4 to construct ordered stem-and-leaf plots, placing the leaves 
on each branch in ascending order, before correctly determining the median and the mode. 

11.8 Box-and- Whisker Plots 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Make and interpret box- and- whisker plots. 

• Analyze effects of outliers. 

• Make box- and- whisker plots using a graphing calculator. 

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Vocabulary 

Terms introduced in this lesson: 

first quartile, third quartile 

five number summary 

whiskers 

inter-quartile range (IQR) 

range 

raw data 
ordered list 

outlier, mild outlier, extreme outlier 

Teaching Strategies and Tips 

Use the introduction to point out that the median can be used to divide a data set into four quarters. See 
also Examples 1 and 2. 

• After finding the quartiles, it is possible to construct the five-number summary and corresponding 
box- and- whisker plot. 

• After finding the quartiles, it is possible to calculate the IQR. 

Emphasize interpreting the box plot: 

• 50% of the data set lies between the first and third quartiles (IQR). 

• 75% of the data set lies above the first quartile; 75% of the data set lies below the third quartile. 

• The range is the distance from one whisker to the other. 

• Compare the relative size of the box to the length of the whiskers: short whiskers indicate clustered 
data; long whiskers indicate a spread-out data set. 

• If one whisker is shorter than another, then the distribution is skewed. 

Construct box-and-whisker plots for two data sets and compare them side-by-side. Point out that this 
makes drawing inferences easy. See Example 3. 

Compare the range and IQR for a data set. Ask: 

• For what kind of distributions should the IQR be used? the range? 

Additional Example: 

For the data set below, calculate the range and IQR. Which measure of dispersion do you think will give a 
better indication of the spread in the data? 

2,5,8,8,8,9,9,10,100 

Error Troubleshooting 

NONE. 

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Chapter 12 

TE Rational Equations and 
Functions; Topics in 
Statistics 



Overview 

In this chapter, students are introduced to inverse variation. They graph rational functions and divide 
polynomials. After being introduced to rational expressions they learn to add, subtract, multiply, and 
divide them. Students then solve rational equations. The chapter ends with surveys and sampling methods. 

Suggested pacing: 

Inverse Variation Models - 1 hr 

Graphs of Rational Functions - 1 - 2 hrs 

Division of Polynomials - 1 hr 

Rational Expressions - 1 hr 

Multiplication and Division of Rational Expressions - 1 hr 

Addition and Subtraction of Rational Expressions - 1 - 2 hrs 

Solutions of Rational Equations - 1 hr 

Surveys and Samples - 1 hr 

If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex- 
Book and the Assessment Answers FlexBook please contact us at teacher-requests@ckl2.org. 

Problem-Solving Strand for Mathematics 

The first portion of this chapter cements connections between algebraic and geometric ways (graphical 
displays) of representing functions and relationships in mathematics. Vertical, horizontal, and oblique 
asymptotes visually confirm what students have learned earlier in their studies: division by zero is unde- 
fined in our number system. The connection between inverse variation models and the graphs of rational 
functions is presented in the examples and review questions which pose real- world problems using rational 
functions. 

In the mid-lesson of this unit, the division of polynomials is related back once again to graphing and the 
horizontal asymptote studied previously. In the lesson, Rational Expressions, values that are excluded when 

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simplifying rational expressions are shown to be those very values that are vertical asymptotes, values that 
cannot exist for x. Students who become adept at moving between the algebraic and the graphic will have 
more insights into problems they encounter and thus more tools for tackling future challenges. 

The last lesson of the unit looks closely at topics in statistics. Students are asked to identify biased questions 
as well as biased sample populations and to display, analyze, and interpret statistical data effectively. Let 
students know they will have a survey project to complete before introducing the Surveys and Samples 
materials lesson and before reviewing Examples 3 and 4, Designing a Survey, and Examples 5 and 6, 
Display, Analyze, and Interpret Data, since the information needs to be seen in context in order for its 
value to be recognized. 

Alignment with the NCTM Process Standards 

The first lessons of this chapter consistently recognize and use connections among mathematical ideas 
(CON.l), encourage students to understand how mathematical ideas interconnect and build on one another 
to produce a coherent whole (CON. 2), and — especially in real-world problem solving scenarios — recognize 
and apply mathematics in contexts outside of mathematics (CON. 3). In the lessons, Rational Expressions 
and Solutions of Rational Equations, classic work and motion problems are presented so as to put these 
problem-solving techniques into meaningful perspective (PS.l, PS. 2, and PS. 3). In the Surveys and Samples 
lesson, students create and use representations to organize, record, and communicate mathematical ideas 
(R.l); select, apply, and translate among mathematical representations to solve problems (R.2); and use 
representations to model and interpret physical, social, and mathematical phenomena (R.3). 

• CON.l - Recognize and use connections among mathematical ideas. 

• CON. 2 - Understand how mathematical ideas interconnect and build on one another to produce a 
coherent whole. 

• CON. 3 - Recognize and apply mathematics in contexts outside of mathematics. 

• PS.l - Build new mathematical knowledge through problem solving. 

• PS. 2 - Solve problems that arise in mathematics and in other contexts. 

• PS. 3 - Apply and adapt a variety of appropriate strategies to solve problems. 

• R.l - Create and use representations to organize, record, and communicate mathematical ideas. 

• R.2 - Select, apply, and translate among mathematical representations to solve problems. 

• R.3 - Use representations to model and interpret physical, social, and mathematical phenomena. 

12.1 Inverse Variation Models 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Distinguish direct and inverse variation. 

• Graph inverse variation equations. 

• Write inverse variation equations. 

• Solve real-world problems using inverse variation equations. 

Vocabulary 

Terms introduced in this lesson: 

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variation 

direct variation 

inverse variation 

joint variation 

constant of proportionality 

increase, decrease 

Teaching Strategies and Tips 

This lesson focuses on inverse variation models and graphing inverse variation equations. Use it to motivate 
rational functions, which are covered in the next six lessons. 

Remind students about having learned direct variation in chapter Graphs of Equations and Functions. 

• Point out that direct variation is a linear relationship. 

• The x and v-intercepts are 0. 

• The slope of the line is the only parameter, denoted by k, and called the constant of proportionality. 

• It takes only one more point to determine the direct variation. 

Some examples of direct variation relationships are: 

• Height of a person and the length of their shadow on flat ground. 

• Circumference and radius of the circle. 

• Weight of an object on a spring and the amount the spring has stretched. 

In the examples and Review Questions, have students decide on a variation model first and then solve for 
the constant of proportionality using the given information. This determines the equation of the variation 
which is necessary for answering the rest of the problem. 

Use Example 1 to illustrate the graph of an inverse variation. 

• Construct a similar table of values. Allow students to observe the function's behavior numerically. 

Remind students of scientific notation in Example 6. 

In applied problems such as Examples 5 and 6, emphasize that direct variations are ubiquitous and signif- 
icant in the real-world. 

In Review Questions 1-4, encourage students to apply stretches to the basic graph y = -. 

Example: 

Graph the following inverse variation relationship. 

10 

y= — ■ 

x 
Hint: Since y = — = 10 • -, the graph can be obtained from that of y = - by stretching by a factor of 10. 

Error Troubleshooting 

Remind students in Example 1 that dividing by zero is undefined. 
Remind students in Example 6 to square the 5.3 which is in parentheses: 

K = 740(5.3 x 10 -11 ) 2 = 740 • 5.3 2 • 10~ 22 
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12.2 Graphs of Rational Functions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Compare graphs of inverse variation equations. 

• Graph rational functions. 

• Solve real-world problems using rational functions. 

Vocabulary 

Terms introduced in this lesson: 

rational function 

horizontal asymptote, vertical asymptote 

oblique (slant) asymptote 

eaching Strategies and Tips 

Reconstruct the tables in Examples 2-4 to remind students of the inverse relationship. 
Explore several rational functions side-by-side. 

• Have students make note of the degrees of the numerator and denominator and any horizontal and 
vertical asymptotes. 

• Point out that what sets rational functions apart from other functions in this course is division. 

• Division creates the asymptotes and branches. 

• Remind students that dividing by zero is undefined and is denoted on the graph by a vertical dashed 
line. 

Asymptotes are denoted by dashed lines. Remind students that asymptotes are not part of the function 
and only serve to show how the graph approaches certain values. 

• Point out that graphing calculators may display asymptotes using a solid line. 

Have students rewrite the steps for finding asymptotes preceding Example 5 for themselves. 
Encourage graphing rational functions by hand. Use a graphing calculator only as a way to check. 

• Sketching graphs can solidify student understanding of ^-intercepts, y-intercepts, factoring, and 
domains. 

Error Troubleshooting 

In Examples 2-4, have students choose enough values for their tables to determine the behavior of the 
function accurately. Remind them to pick values close to the vertical asymptotes. 

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12.3 Division of Polynomials 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Divide a polynomial by a monomial. 

• Divide a polynomial by a binomial. 

• Rewrite and graph rational functions. 

Vocabulary 

Terms introduced in this lesson: 

rational expression 

numerator 

denominator 

common denominator 

dividend 

divisor 

quotient 

remainder 

Teaching Strategies and Tips 

Students learned in chapter Factoring Polynomials how to add, subtract, and multiply polynomials. This 
lesson completes that discussion with dividing polynomials. 

• Emphasize that the quotient of two polynomials forms a rational expression which is studied in its 
own right (rational functions). 

Use Example 1 to demonstrate dividing a polynomial by a monomial. 

• Remind students that each term in the numerator must be divided by the monomial in the denomi- 
nator. See Example 2. 

Use Example 3 to motivate long division of polynomials. 

• To write the answer, remind students that: 

dividend remainder 

— = quotient -\ 

divisor divisor 

• To check an answer, have students use the equivalent form: 

dividend = (divisor X quotient) + remainder 

Have students rewrite for themselves the four cases for graphing rational functions preceding Example 5. 
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Error Troubleshooting 

General Tip: Students often incorrectly cancel a factor not common to all the terms. 

• Example: 

dx + b x + b 



& y 

• When students cancel the a above, they violate order of operations. Remind students that the 
fraction sign is a grouping symbol (parentheses) and therefore the numerator and denominator must 
be simplified before dividing. 

• Otherwise, if the numerator and denominator are completely factored, then the order of operations 
says to multiply or divide; therefore, canceling is justified. 

• Have students write out the step preceding the canceling: 

Example: 

ax + ab a{x + b) 
ay ay 

Then canceling is apparent: 

ax + ab a(x + b) d(x + b) x + b 



ay ay </ry y 

• Other common cancelling errors are: 

a. / 1~" + i+O- (forgetting to remove the canceled factor) 



x+fi x 

12 A Rational Expressions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Simplify rational expressions. 

• Find excluded values of rational expressions. 

• Simplify rational models of real-world situations. 

Vocabulary 

Terms introduced in this lesson: 

lowest terms 

canceling common factors 
common terms 
excluded value 
removable zero 



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Teaching Strategies and Tips 

In this lesson, students learn what removable zeros are, how to find them, how they are related to excluded 
values, and what they look like graphically. 

• Point out that removable zeros are "divisions by zero" removed by simplifying the rational expression. 

• See Example 2. 

Have students review factoring. 
Additional Examples: 



a. 



Ifi.v 



m = 4x(j ± 3) =4x + 3) (x ^ Q) 



Ax Ax 

,2 



t 56a- 3 + 14* 2 _ 14.r 2 (4*+l) _ 14^2 / . _1\ 
U - 4.v 2 +5x+l (x+l)(4x+l) jc+1' \ Xr A> 

Optional: In some rational expressions, it may appear to students like nothing will cancel at first, despite 
the similar terms. Suggest that students try factoring out a negative and then rearranging the order of the 
terms. 

L. x _ x x _ l 1, X T- 1 

When factoring out the minus sign, suggest that students use brackets to keep the negative outside and so 
that it will be harder to lose it. 

~(x-l)(x+l)' 



j 1-x 2 __ (!-*)(!+*) 

U - x 2 +x-2 ~ (x-l)(x+2) 



(x-l)(x+2) 



x+2 > •* ^ L 



Point out that the rules for working with rational expressions are the same as those for working with 
ordinary fractions. 

• Simplifying a rational expression means the same as simplifying a fraction - that the numerator and 
denominator of the rational expression have no common factors. 

• Remind students after canceling all the terms of a numerator (or denominator) that a factor of 1 
remains. 

In Example 3, have students use scientific notation. 

Error Troubleshooting 

General Tip: Suggest that students check their work after simplifying a rational expression by substituting 
the variable with a number. 

General Tip: Remind students that in order to cancel a factor, it must be common to the entire numerator 
and denominator. 

12.5 Multiplication and Division of Rational Ex- 
pressions 

Learning Objectives 

At the end of this lesson, students will be able to: 
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• Multiply rational expressions involving monomials. 

• Multiply rational expressions involving polynomials. 

• Multiply a rational expression by a polynomial. 

• Divide rational expressions involving polynomials. 

• Divide a rational expression by a polynomial. 

• Solve real- world problems involving multiplication and division of rational expressions. 

Vocabulary 

NONE 

Teaching Strategies and Tips 

Use the ordinary fractions in Example 1 to motivate multiplying and dividing variable rational expressions. 

Use Example 2 to point out that an answer can be more readily obtained by reducing common factors 
before multiplying. 

• Suggest students to factor all polynomial expressions as much as possible first. 

Error Troubleshooting 

In Review Questions 9, 12, and 16, remind students to write the expression on the right as a fraction with 
denominator equal to 1, and then proceed to multiply or divide. 

General Tip: Remind students to rewrite division problems as multiplication problems. 

General Tip: In a division problem, some students will invert the first fraction. Remind students to invert 
the second. 

12.6 Addition and Subtraction of Rational Ex- 
pressions 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Add and subtract rational expressions with the same denominator. 

• Find the least common denominator of rational expressions. 

• Add and subtract rational expressions with different denominators. 

• Solve real- world problems involving addition and subtraction of rational expressions. 

Vocabulary 

Terms introduced in this lesson: 

least common denominator (LCD) 

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least common multiple (LCM) 
prime factorization 
factor completely 
equivalent fraction 

Teaching Strategies and Tips 

Use the ordinary fractions in Examples 1 and 5 to motivate adding and subtracting variable rational 
expressions with and without common denominators, respectively. 

Draw the analogy between finding the LCM of polynomials and the LCM of integers. 

• Use prime factorization of numbers and polynomials. 

• In general, the LCM is found by taking each factor to the highest power that it appears in each 
expression. 

In Examples 6, 8, and 9, remind students to distribute the minus to each term in the second rational 
expression. See also Review Questions 4, 5, 8, 17, 18, 21, 23, 27, and 30. 

Draw the analogy between the formula: part of the task completed = rate of work ■ time spent on the task 
and the formula: distance = rate ■ time. 

In Review Questions 34-36, encourage students to set up a table similar to the one in Example 12. 

• Emphasize that all known and unknown variables for each person or machine can be listed, which 
makes organizing the given information easy. 

• Combining parts of the task completed by each person or machine is a matter of reading across the 
rows or down the columns depending on how the table is set up. 

• Many students find tables useful for work problems; others rely heavily upon it. 

In Review Questions 34-36, 

• Suggest that students begin by looking at the part of the task completed by each person or machine 
separately. 

• Encourage students to check the reasonableness of their answers. Ask: What kind of answer should 
we expect based upon the given information? 

Error Troubleshooting 

In Review Questions 7 and 8, suggest that students factor out a negative from the second rational expression 
first. 

• In Example 7, the LCM of x - 4 and 4 - x = -(x - 4) is x - 4. 

General Tip: Remind students to find the LCD of rational expressions by factoring. Students needlessly 
use larger common multiples when expressions are not completely factored. 

General Tip: Have students leave the LCM in factored form. This makes simplifying and determining 
excluded values easier. 

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12.7 Solutions of Rational Equations 

Learning Objectives 

At the end of this lesson, students will be able to: 



Solve rational equations using cross products. 

Solve rational equations using lowest common denominators. 

Solve real- world problems with rational equations. 



Vocabulary 

Terms introduced in this lesson: 

rational equation 
cross products 

Teaching Strategies and Tips 

Students should now be able to solve linear, quadratic, and radical equations; and be able to graph linear, 
quadratic, exponential, and radical functions. In this lesson, rational equations and functions are added 
to these lists. 

Emphasize that the first step in solving rational equations is eliminating the denominators on all the terms. 

• Emphasize that the last step is checking their answers in the original equation. 
Use Example 1 to remind students about cross-multiplication. 

• Suggest that students simplify the terms of the equation and move them to one side before cross- 
multiplying. 

Use Example 4 to motivate using the LCD strategy. 

• Students learn to eliminate denominators by multiplying all terms by the LCD. 

• Emphasize that this method is preferred over cross-multiplication when there are two or more terms 
in an equation. 

• Have students completely factor each expression first. 

After multiplying both sides by the LCD, have students use colors to cancel like factors. This helps students 
track their canceling and prevents canceling a factor more than once. 

Reconstruct the tables in Examples 7-9 as a useful way to display the given information in Review Questions 
19-24. 

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Error Troubleshooting 

General Tip: Remind students to check their answers in the original equation. 

• Emphasize that this is a necessary step for rational equations since the variable appears in denomi- 
nators. 

• Essentially, if an answer makes any denominator zero, then that value is not a solution to the equation. 

12.8 Surveys and Samples 

Learning Objectives 

At the end of this lesson, students will be able to: 

• Classify sampling methods. 

• Identify biased samples. 

• Identify biased questions. 

• Design and conduct a survey. 

• Display, analyze, and interpret statistical survey data. 

Vocabulary 

Terms introduced in this lesson: 

census 

sampling method 

representative sample 

random sampling 

stratified sampling 

sample size 

bias 

biased sample 

population 

sample 

cherry picking 

non-response bias 

self-selected respondents 

poll 

survey 

biased question 

question order 

design and conduct a survey 

face-to-face interview 

self-administered survey 

survey data 

Teaching Strategies and Tips 

Students learn about surveys and samples as ways of collecting information: 
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• Examples accompany each type of method and help students put them into context. 

• Emphasize the advantages and disadvantages of each type of sampling method: census, random, and 
stratified. 

Additional Examples: 

Identify the type of sampling used. 

a. A superintendent has a computer generate a list of 100 teachers from the 10 schools in her district and 
interviews them about working conditions. 

Answer: Random sample 

b. At a certain high school there are 730 freshmen, 512 sophomores, 475 juniors, and 103 seniors. A 
reporter from the school newspaper interviews 15 freshmen, 12 sophomores, 13 juniors, and 10 seniors 
about their college plans. 

Answer: Stratified sampling 

c. Around mid-semester, a certain university requires a teacher's evaluation. A form is given to each 
student in every class to fill out. 

Answer: Census 

Emphasize that samples consisting of over or under-represented sub-groups are biased. 

• Point out that a biased sample does not accurately reflect the spread of people present in the popu- 
lation and, therefore, the sample should not be expected to represent the entire population. 

Additional Examples: 

Identify each sample as biased or unbiased. If the sample is biased explain how you would improve your 
sampling method. 

a. County health officials randomly selected non-smokers walking out of a bar. They politely asked every 
other person whether or not second-hand smoke affects them. 

Hint: Possible under-represented group; people who dislike second-hand smoke tend to stay away from 
bars. 

b. At a popular ski resort, people were surveyed about the cafeteria food. The survey was handed out in 
the cafeteria and was to be mailed in, postage paid. The results were to be published the county newspaper. 

Hint: Possible bias includes the voluntary response of the people being surveyed. For example, those who 
find the time and effort to mail in the survey may be predisposed to a certain opinion. 

c. A potato chip manufacturer packaging bags of chips maintains quality control by randomly selecting 
100 bags over the course of 48 hours and weighs each bag. They then inspect 10 bags by opening them 
and tasting the chips. 

Hint: No apparent bias. 

d. People visiting the website of a popular teen magazine had the option of participating in a poll on 
whether the legal drinking age should be lowered. 

Hint: Possible bias includes the voluntary response of the people being surveyed. 

Teachers are encouraged to read and discuss with their students the ways to spot biased questions. See 
the list following Example 2. 

Additional Examples: 

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Examine the question for possible bias. If you think the question is biased, indicate how to propose a better 
question. 

a. Given that underage drinking is responsible for 17% of all car accidents, do you think the legal drinking 
age should be lowered? 

Hint: Biased. Possible alternative: "Do you think lowering the legal drinking age is appropriate?" 

b. Should corporations that use diesel fuel in their transports, and thus pollute the environment, pay an 
environmental tax to help clean the air? 

Hint: Biased. Possible alternative: "Should corporations using diesel fuel pay an environmental tax?" 

c. Are you in favor of continued funding for the high school's remarkable theater program? 

Hint: Biased. Possible alternative: "Do you favor continued funding for the high school theater program?" 

d. Do you think all high school students should be required to take a physical education course? 
Hint: Likely unbiased. 

Use Examples 3 and 4 to demonstrate step-by-step how to conduct a survey. 

Use the last part of the lesson to show students how to display, analyze, and interpret survey data. 
Appropriate ways to display data are: pie-charts and bar-graphs for categorical data; stem-and-leaf plots, 
histograms, and box- and- whisker plots for numerical data. 

Error Troubleshooting 

NONE 



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